Precalculus Quiz: Sine And Cosine Of Complementary Angles
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Sine And Cosine Of Complementary AnglesQuestion 1 of 20

In right triangle ABCABC with ∠C=90∘\angle C=90^\circ, the acute angles ∠A\angle A and ∠B\angle B are complementary. How does sin⁡(A)\sin(A) relate to cos⁡(B)\cos(B)?​​

sin⁡(A)=cos⁡(B)\sin(A)=\cos(B)
sin⁡(A)=sin⁡(B)\sin(A)=\sin(B)
sin⁡(A)=cos⁡(180∘−B)\sin(A)=\cos(180^\circ-B)
sin⁡(A)=sec⁡(B)\sin(A)=\sec(B)
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Precalculus Quiz

Precalculus Quiz: Sine And Cosine Of Complementary Angles

Practice Sine And Cosine Of Complementary Angles in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Sine And Cosine Of Complementary Angles, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

In right triangle ABCABC with ∠C=90∘\angle C=90^\circ, the acute angles ∠A\angle A and ∠B\angle B are complementary. How does sin⁡(A)\sin(A) relate to cos⁡(B)\cos(B)?​​

  1. sin⁡(A)=cos⁡(B)\sin(A)=\cos(B) (correct answer)
  2. sin⁡(A)=sin⁡(B)\sin(A)=\sin(B)
  3. sin⁡(A)=cos⁡(180∘−B)\sin(A)=\cos(180^\circ-B)
  4. sin⁡(A)=sec⁡(B)\sin(A)=\sec(B)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles A and B, where A + B = 90°, consider a side that is opposite to angle A (making sin(A) = opposite/hypotenuse). That same side is adjacent to angle B (making cos(B) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(A) = cos(B). Choice A is correct because it correctly states that sin(A) = cos(B) when angles A and B are complementary in a right triangle. Choice B incorrectly claims sin(A) = sin(B), which would only be true if the angles were equal, not complementary. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 2

Compare sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). (Note that 20∘20^\circ and 70∘70^\circ are complementary.) Which statement is true?

  1. sin⁡(20∘)>cos⁡(70∘)\sin(20^\circ)>\cos(70^\circ)
  2. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ)=\cos(70^\circ) (correct answer)
  3. sin⁡(20∘)<cos⁡(70∘)\sin(20^\circ)<\cos(70^\circ)
  4. No relationship can be determined from complementarity.
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice B is correct because it correctly states that sin(20°) = cos(70°) based on the complementary relationship. Choice A claims sin(20°) > cos(70°), but since they are equal, this is incorrect. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 3

In a right triangle, the acute angles α\alpha and β\beta are complementary (α+β=90∘\alpha+\beta=90^\circ). Which statement must be true?​

  1. cos⁡(α)=cos⁡(90∘−β)\cos(\alpha)=\cos(90^\circ-\beta)
  2. cos⁡(α)=sin⁡(β)\cos(\alpha)=\sin(\beta) (correct answer)
  3. cos⁡(α)=sin⁡(180∘−β)\cos(\alpha)=\sin(180^\circ-\beta)
  4. cos⁡(α)=csc⁡(β)\cos(\alpha)=\csc(\beta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles α and β, where α + β = 90°, consider a side that is opposite to angle α (making sin(α) = opposite/hypotenuse); that same side is adjacent to angle β (making cos(β) = adjacent/hypotenuse), but here we see cos(α) = sin(β). Choice B is correct because it correctly applies cos(α) = sin(β) based on the complementary relationship. Choice D confuses the complementary angle relationship with the reciprocal relationship, using cosecant or secant instead of the complementary angle's cosine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 4

In the right triangle, the acute angles are complementary. If one acute angle is 25∘25^\circ, which statement correctly uses the complementary angle relationship to rewrite sin⁡(25∘)\sin(25^\circ)?

  1. sin⁡(25∘)=cos⁡(65∘)\sin(25^\circ)=\cos(65^\circ) (correct answer)
  2. sin⁡(25∘)=cos⁡(155∘)\sin(25^\circ)=\cos(155^\circ)
  3. sin⁡(25∘)=sin⁡(65∘)\sin(25^\circ)=\sin(65^\circ)
  4. sin⁡(25∘)=cos⁡(25∘)\sin(25^\circ)=\cos(25^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since 25° and 65° are complementary (they sum to 90°), we know that sin(25°) = cos(65°). Choice A is correct because it correctly applies sin(25°) = cos(65°) based on the complementary relationship in the right triangle. Choice B confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 5

Which equation correctly expresses the complementary angle relationship for degrees?​​

  1. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  2. cos⁡(θ)=cos⁡(90∘−θ)\cos(\theta)=\cos(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta) (correct answer)
  4. sin⁡(θ)=cos⁡(180∘−θ)\sin(\theta)=\cos(180^\circ-\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). Choice C is correct because it correctly states the cofunction relationship sin(θ) = cos(90° - θ) for complementary angles. Choice D confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 6

In the right triangle, one acute angle measures 35∘35^\circ. What is the measure of the angle complementary to 35∘35^\circ?​

  1. 145∘145^\circ
  2. 55∘55^\circ (correct answer)
  3. 45∘45^\circ
  4. 125∘125^\circ
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 35° and the other acute angle are complementary (they sum to 90°), we can immediately conclude that the other angle is 90° - 35° = 55°. Choice B is correct because it correctly identifies the complementary angle as 55° by subtracting from 90°. Choice A confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - 35° instead of 90° - 35°. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 7

Compare the two expressions sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). Given that 20∘20^\circ and 70∘70^\circ are complementary angles, which statement is correct?​

  1. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ)=\cos(70^\circ) (correct answer)
  2. sin⁡(20∘)=sin⁡(70∘)\sin(20^\circ)=\sin(70^\circ)
  3. sin⁡(20∘)=cos⁡(110∘)\sin(20^\circ)=\cos(110^\circ)
  4. sin⁡(20∘)=sec⁡(70∘)\sin(20^\circ)=\sec(70^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice A is correct because it correctly applies sin(20°) = cos(70°) using the complementary relationship. Choice B claims sin(20°) = sin(70°), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 8

Compare sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). Given that 20∘20^\circ and 70∘70^\circ are complementary, which statement is true?​​​

  1. sin⁡(20∘)<cos⁡(70∘)\sin(20^\circ) < \cos(70^\circ)
  2. sin⁡(20∘)>cos⁡(70∘)\sin(20^\circ) > \cos(70^\circ)
  3. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ)=\cos(70^\circ) (correct answer)
  4. sin⁡(20∘)=sin⁡(70∘)\sin(20^\circ)=\sin(70^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice C is correct because it correctly states that sin(20°) = cos(70°) using the cofunction relationship. Choice D claims sin(20°) = sin(70°), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 9

In the right triangle, one acute angle measures 35∘35^\circ. The other acute angle is complementary to it. What is the measure of that complementary angle?

  1. 55∘55^\circ (correct answer)
  2. 145∘145^\circ
  3. 125∘125^\circ
  4. 45∘45^\circ
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 35° and 55° are complementary (they sum to 90°), we know that sin(35°) = cos(55°). Given that one angle is 35°, we can immediately conclude that the other is 55°. Choice A is correct because it correctly identifies complementary angles sum to 90°. Choice B confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 10

In the right triangle, one acute angle measures 35∘35^\circ. What is the measure of the angle complementary to 35∘35^\circ?

  1. 145∘145^\circ
  2. 55∘55^\circ (correct answer)
  3. 45∘45^\circ
  4. 125∘125^\circ
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 35° and the other acute angle are complementary (they sum to 90°), we can immediately conclude that the other angle is 90° - 35° = 55°. Choice B is correct because it correctly identifies the complementary angle as 55° by subtracting from 90°. Choice A confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - 35° instead of 90° - 35°. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 11

Compare sin⁡(20∘)\sin(20^\circ) and cos⁡(70∘)\cos(70^\circ). Given that 20∘20^\circ and 70∘70^\circ are complementary, which statement is true?

  1. sin⁡(20∘)<cos⁡(70∘)\sin(20^\circ) < \cos(70^\circ)
  2. sin⁡(20∘)>cos⁡(70∘)\sin(20^\circ) > \cos(70^\circ)
  3. sin⁡(20∘)=cos⁡(70∘)\sin(20^\circ) = \cos(70^\circ) (correct answer)
  4. No relationship can be determined from complementarity alone.
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 20° and 70° are complementary (they sum to 90°), we know that sin(20°) = cos(70°). Choice C is correct because it correctly identifies complementary angles sum to 90° and applies sin(θ) = cos(90° - θ). Choice D claims no relationship can be determined from complementarity alone, but the cofunction identity directly shows equality. Key to complementary angles: remember that two angles are complementary if they sum to 90°, and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 12

In a right triangle, angle A=25∘A=25^\circ and angle B=65∘B=65^\circ, so AA and BB are complementary. Which equation correctly connects cos⁡(A)\cos(A) and sin⁡(B)\sin(B)?

  1. cos⁡(25∘)=sin⁡(65∘)\cos(25^\circ)=\sin(65^\circ) (correct answer)
  2. cos⁡(25∘)=cos⁡(65∘)\cos(25^\circ)=\cos(65^\circ)
  3. cos⁡(25∘)=sin⁡(115∘)\cos(25^\circ)=\sin(115^\circ)
  4. cos⁡(25∘)=sin⁡(25∘)\cos(25^\circ)=\sin(25^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since A and B are complementary (they sum to 90° or π/2), we know that cos(A) = sin(B). Choice A is correct because it correctly applies cos(25°) = sin(65°) using the complementary relationship. Choice B claims cos(25°) = cos(65°), using the same function for both angles, when the cofunction relationship requires switching from cosine to sine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 13

In a right triangle, the acute angles α\alpha and β\beta are complementary (α+β=90∘\alpha+\beta=90^\circ). Which statement must be true?

  1. cos⁡(α)=cos⁡(90∘−β)\cos(\alpha)=\cos(90^\circ-\beta)
  2. cos⁡(α)=sin⁡(β)\cos(\alpha)=\sin(\beta) (correct answer)
  3. cos⁡(α)=sin⁡(180∘−β)\cos(\alpha)=\sin(180^\circ-\beta)
  4. cos⁡(α)=csc⁡(β)\cos(\alpha)=\csc(\beta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles α and β, where α + β = 90°, consider a side that is opposite to angle α (making sin(α) = opposite/hypotenuse); that same side is adjacent to angle β (making cos(β) = adjacent/hypotenuse), but here we see cos(α) = sin(β). Choice B is correct because it correctly applies cos(α) = sin(β) based on the complementary relationship. Choice D confuses the complementary angle relationship with the reciprocal relationship, using cosecant or secant instead of the complementary angle's cosine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine.

Question 14

In a right triangle, angle A=25∘A=25^\circ and angle B=65∘B=65^\circ, so AA and BB are complementary. Which equation correctly connects cos⁡(A)\cos(A) and sin⁡(B)\sin(B)?​

  1. cos⁡(25∘)=sin⁡(65∘)\cos(25^\circ)=\sin(65^\circ) (correct answer)
  2. cos⁡(25∘)=cos⁡(65∘)\cos(25^\circ)=\cos(65^\circ)
  3. cos⁡(25∘)=sin⁡(115∘)\cos(25^\circ)=\sin(115^\circ)
  4. cos⁡(25∘)=sin⁡(25∘)\cos(25^\circ)=\sin(25^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since A and B are complementary (they sum to 90° or π/2), we know that cos(A) = sin(B). Choice A is correct because it correctly applies cos(25°) = sin(65°) using the complementary relationship. Choice B claims cos(25°) = cos(65°), using the same function for both angles, when the cofunction relationship requires switching from cosine to sine. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement). Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 15

Given complementary angles θ\theta and 90∘−θ90^\circ-\theta, which equation correctly expresses the cofunction identity?

  1. cos⁡(θ)=sin⁡(90∘−θ)\cos(\theta)=\sin(90^\circ-\theta) (correct answer)
  2. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(180∘−θ)\sin(\theta)=\cos(180^\circ-\theta)
  4. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. The cofunction relationship between sine and cosine—that sin(θ) = cos(90° - θ)—derives from the fact that in a right triangle, the side that is opposite to one acute angle is adjacent to the complementary acute angle. The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). This can be rewritten as cos(θ) = sin(90° - θ), which is the form shown in choice A. Choice A is correct because it correctly states the cofunction identity cos(θ) = sin(90° - θ), which is equivalent to the standard form sin(θ) = cos(90° - θ). Choice B incorrectly claims sin(θ) = sin(90° - θ), using the same function for both angles, when the cofunction relationship requires switching from sine to cosine. To verify the complementary angle relationship on the unit circle, observe that as you move from angle θ to angle 90° - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(90° - θ).

Question 16

In a right triangle, the two acute angles are complementary. If one acute angle is 65∘65^\circ, which expression equals sin⁡(65∘)\sin(65^\circ) by the complementary angle relationship?

  1. cos⁡(25∘)\cos(25^\circ) (correct answer)
  2. cos⁡(115∘)\cos(115^\circ)
  3. sin⁡(25∘)\sin(25^\circ)
  4. cos⁡(65∘)\cos(65^\circ)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. Since the two acute angles are complementary and one is 65°, the other must be 90° - 65° = 25°. By the complementary angle relationship, sin(65°) = cos(90° - 65°) = cos(25°). Choice A is correct because it correctly identifies that sin(65°) = cos(25°) when 65° and 25° are complementary angles in a right triangle. Choice D incorrectly suggests cos(65°), which would equal sin(25°), not sin(65°). In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 17

Using the cofunction identity for complementary angles, which equation correctly expresses the relationship between sine and cosine?​

  1. sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta)=\cos(90^\circ-\theta) (correct answer)
  2. sin⁡(θ)=sin⁡(90∘−θ)\sin(\theta)=\sin(90^\circ-\theta)
  3. sin⁡(θ)=cos⁡(180∘−θ)\sin(\theta)=\cos(180^\circ-\theta)
  4. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. The cofunction relationship between sine and cosine—that sin(θ) = cos(90° - θ)—derives from the fact that in a right triangle, the side that is opposite to one acute angle is adjacent to the complementary acute angle. The equation sin(θ) = cos(90° - θ) holds for all angles θ, meaning that if we know the sine of any angle, we automatically know the cosine of its complement (the angle that when added to θ gives 90°). Choice A is correct because it correctly states the cofunction relationship with sin(θ) = cos(90° - θ). Choice C confuses complementary angles (sum to 90°) with supplementary angles (sum to 180°), using 180° - θ instead of 90° - θ. Don't confuse complementary (sum to 90°) with supplementary (sum to 180°)—for supplementary angles, there is no simple cofunction relationship between sine and cosine. Key to complementary angles: remember that two angles are complementary if they sum to 90° (or π/2), and the cofunction relationship sin(θ) = cos(90° - θ) allows you to convert between sine and cosine using the complement.

Question 18

In a right triangle, the acute angles are α\alpha and β\beta, and α+β=90∘\alpha+\beta=90^\circ. Which statement must be true?

  1. sin⁡(α)=sin⁡(β)\sin(\alpha)=\sin(\beta)
  2. cos⁡(α)=cos⁡(β)\cos(\alpha)=\cos(\beta)
  3. sin⁡(α)=cos⁡(β)\sin(\alpha)=\cos(\beta) (correct answer)
  4. sin⁡(α)=cos⁡(180∘−β)\sin(\alpha)=\cos(180^\circ-\beta)
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. In a right triangle, the two acute angles are always complementary (they sum to 90°), and by similarity, the side ratios reveal that the sine of one acute angle equals the cosine of the other acute angle. In a right triangle with acute angles α and β, where α + β = 90°, consider a side that is opposite to angle α (making sin(α) = opposite/hypotenuse). That same side is adjacent to angle β (making cos(β) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(α) = cos(β). Choice C is correct because it correctly states that sin(α) = cos(β) when angles α and β are complementary in a right triangle. Choice A incorrectly claims sin(α) = sin(β), which would only be true if the angles were equal, not complementary. In right triangles, always remember that the two acute angles are complementary, so sin of one angle equals cos of the other angle—this is why cosine starts with 'co' (for complement).

Question 19

Using the complementary angle relationship, if sin⁡(30∘)=12\sin(30^\circ)=\tfrac{1}{2}, what is cos⁡(60∘)\cos(60^\circ)?

  1. 32\tfrac{\sqrt{3}}{2}
  2. 22\tfrac{\sqrt{2}}{2}
  3. 12\tfrac{1}{2} (correct answer)
  4. 11
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90° (or π/2 radians), and there is a fundamental relationship: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ) for any angle θ. Since 30° and 60° are complementary (they sum to 90°), we know that sin(30°) = cos(60°). Given that sin(30°) = 1/2, we can immediately conclude that cos(60°) = 1/2. Choice C is correct because it correctly applies the complementary angle relationship to find that cos(60°) = sin(30°) = 1/2. Choice A gives √3/2, which is actually cos(30°) or sin(60°), confusing which angle pairs with which value. For special angles, use the complementary pairs: 30° and 60° (or π/6 and π/3) are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2.

Question 20

In a right triangle, one acute angle is 45∘45^\circ, making the other acute angle also 45∘45^\circ (they are complementary). Using the complementary angle relationship, what is cos⁡(45∘)\cos(45^\circ) if sin⁡(45∘)=22\sin(45^\circ)=\tfrac{\sqrt{2}}{2}?

  1. 12\tfrac{1}{2}
  2. 32\tfrac{\sqrt{3}}{2}
  3. 22\tfrac{\sqrt{2}}{2} (correct answer)
  4. 00
Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals 90∘90^\circ (or π/2\pi/2 radians), and there is a fundamental relationship: sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta) = \cos(90^\circ - \theta) and cos⁡(θ)=sin⁡(90∘−θ)\cos(\theta) = \sin(90^\circ - \theta) for any angle θ\theta. For the complementary angles both at 45∘45^\circ (since 45∘+45∘=90∘45^\circ + 45^\circ = 90^\circ), we apply the relationship: sin⁡(45∘)=cos⁡(45∘)\sin(45^\circ) = \cos(45^\circ) and both equal 22\tfrac{\sqrt{2}}{2}, demonstrating the cofunction property with exact values. Choice C is correct because it correctly states cos⁡(45∘)=22\cos(45^\circ) = \tfrac{\sqrt{2}}{2} using the given sin⁡(45∘)=22\sin(45^\circ) = \tfrac{\sqrt{2}}{2} and the complementary relationship. Choice B uses the wrong special triangle value, confusing the 30-60-90 ratios with the 45-45-90 ratios. For special angles, use the complementary pairs: 30∘30^\circ and 60∘60^\circ (or π/6\pi/6 and π/3\pi/3) are complementary, so sin⁡(30∘)=cos⁡(60∘)=12\sin(30^\circ) = \cos(60^\circ) = \tfrac{1}{2} and sin⁡(60∘)=cos⁡(30∘)=32\sin(60^\circ) = \cos(30^\circ) = \tfrac{\sqrt{3}}{2}, but for 45∘45^\circ, it is its own complement.