Precalculus Quiz: Sine And Cosine Of Complementary Angles

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.

Question 1

1. $\sin(20^\circ)>\cos(70^\circ)$
2. $\sin(20^\circ)=\cos(70^\circ)$ (correct answer)
3. $\sin(20^\circ)<\cos(70^\circ)$
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 2

1. $\frac{4}{5}$
2. $\frac{3}{5}$ (correct answer)
3. $\frac{5}{4}$
4. $\frac{5}{3}$

Explanation: The other acute angle measures $90° - 38° = 52°$. Since the two acute angles in a right triangle are complementary, $\cos(52°) = \cos(90° - 38°) = \sin(38°) = \frac{3}{5}$. Choice A gives $\cos(38°)$ instead. Choices C and D are impossible since cosine values must be between -1 and 1.

Question 3

1. 1.0
2. 0.8
3. 0.4
4. 0.6 (correct answer)

Explanation: When you encounter problems involving angles and trigonometric ratios, pay careful attention to which angle is being referenced. This question tests your understanding of complementary angles and the relationship between sine and cosine. The key insight is recognizing that the angle the ladder makes with the wall is complementary to the angle $\theta$ that the ladder makes with the ground. Since these two angles must sum to $90°$, if we call the angle with the wall $\alpha$, then $\alpha + \theta = 90°$, which means $\alpha = 90° - \theta$. Given that $\sin \alpha = 0.6$, we can use the cofunction identity: $\sin(90° - \theta) = \cos \theta$. Therefore, $\cos \theta = 0.6$. Looking at the wrong answers: Choice (A) 1.0 would only be correct if $\theta = 0°$, meaning the ladder lies flat on the ground, which contradicts the setup. Choice (B) 0.8 might tempt you if you incorrectly think $\cos \theta$ should equal $\sqrt{1 - \sin^2 \theta}$ where you mistakenly use $\sin \theta = 0.6$ instead of recognizing that 0.6 is the sine of the complementary angle. Choice (C) 0.4 has no clear mathematical relationship to the given information and likely represents a calculation error. The correct answer is (D) 0.6. Study tip: Remember that sine and cosine are cofunctions, meaning $\sin \alpha = \cos(90° - \alpha)$. When dealing with right triangles, always identify which angle you're working with and use complementary angle relationships when needed.

Question 4

1. $\frac{17}{13}$ (correct answer)
2. $\frac{10}{13}$
3. $\frac{24}{13}$
4. $\frac{7}{13}$

Explanation: Since angles A and B are complementary in a right triangle, we have $\sin B = \cos A = \frac{12}{13}$ and $\cos B = \sin A = \frac{5}{13}$. Therefore, $\sin B + \cos B = \frac{12}{13} + \frac{5}{13} = \frac{17}{13}$. Choice B incorrectly uses $\sin A + \sin B$. Choice C incorrectly uses $\cos A + \cos B$. Choice D results from subtracting instead of adding.

Question 5

1. $\sin(\theta)=\cos(90^\circ-\theta)$ (correct answer)
2. $\sin(\theta)=\sin(90^\circ-\theta)$
3. $\sin(\theta)=\cos(180^\circ-\theta)$
4. $\sin(\theta)=\cos(\theta)$

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 6

1. $\sin(\alpha)=\sin(\beta)$
2. $\cos(\alpha)=\cos(\beta)$
3. $\sin(\alpha)=\cos(\beta)$ (correct answer)
4. $\sin(\alpha)=\cos(180^\circ-\beta)$

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 7

1. $\tfrac{\sqrt{3}}{2}$
2. $\tfrac{\sqrt{2}}{2}$
3. $\tfrac{1}{2}$ (correct answer)
4. $1$

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 8

1. $\tfrac{1}{2}$
2. $\tfrac{\sqrt{3}}{2}$
3. $\tfrac{\sqrt{2}}{2}$ (correct answer)
4. $0$

Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. Two angles are complementary if their sum equals $90^\circ$ (or $\pi/2$ radians), and there is a fundamental relationship: $\sin(\theta) = \cos(90^\circ - \theta)$ and $\cos(\theta) = \sin(90^\circ - \theta)$ for any angle $\theta$. For the complementary angles both at $45^\circ$ (since $45^\circ + 45^\circ = 90^\circ$), we apply the relationship: $\sin(45^\circ) = \cos(45^\circ)$ and both equal $\tfrac{\sqrt{2}}{2}$, demonstrating the cofunction property with exact values. Choice C is correct because it correctly states $\cos(45^\circ) = \tfrac{\sqrt{2}}{2}$ using the given $\sin(45^\circ) = \tfrac{\sqrt{2}}{2}$ 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: $30^\circ$ and $60^\circ$ (or $\pi/6$ and $\pi/3$) are complementary, so $\sin(30^\circ) = \cos(60^\circ) = \tfrac{1}{2}$ and $\sin(60^\circ) = \cos(30^\circ) = \tfrac{\sqrt{3}}{2}$, but for $45^\circ$, it is its own complement.

Question 9

1. $\tfrac{\sqrt{3}}{2}$
2. $\tfrac{1}{2}$ (correct answer)
3. $-\tfrac{1}{2}$
4. $\tfrac{\sqrt{2}}{2}$

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. For the complementary angles π/6 and π/3, we apply the relationship: sin(π/6) = cos(π/3) and both equal 1/2, while sin(π/3) = cos(π/6) and both equal √3/2, demonstrating the cofunction property with exact values. Choice B is correct because it correctly states cos(π/3) = 1/2 using the given sin(π/6) = 1/2 and the complementary relationship. Choice A 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° 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 10

1. Because $40^\circ+50^\circ=90^\circ$, $\sin(40^\circ)=\cos(50^\circ)$. (correct answer)
2. Because $40^\circ+50^\circ=180^\circ$, $\sin(40^\circ)=\cos(50^\circ)$.
3. Because $\sin^2(40^\circ)+\cos^2(50^\circ)=1$, $\sin(40^\circ)=\cos(50^\circ)$.
4. Because sine and cosine are always equal for acute angles, $\sin(40^\circ)=\cos(50^\circ)$.

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. 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 identifies complementary angles sum to 90° and applies the cofunction relationship. 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. 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.

Question 11

1. $\sin(A)=\sin(B)$
2. $\sin(A)=\cos(B)$ (correct answer)
3. $\sin(A)=\cos(180^\circ-B)$
4. $\sin(A)=\sec(B)$

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 angle A = 25° and angle C = 90° in the right triangle, angle B must equal 65° (because 25° + 65° + 90° = 180°), making angles A and B complementary (25° + 65° = 90°). By the complementary angle relationship, sin(A) = sin(25°) = cos(90° - 25°) = cos(65°) = cos(B). Choice B is correct because it correctly identifies that sin(A) = cos(B) when A and B are complementary angles. Choice A incorrectly claims sin(A) = sin(B), which would only be true if A = B, but here A = 25° and B = 65°. 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 12

1. $\sin(65^\circ)$ (correct answer)
2. $\cos(65^\circ)$
3. $\sin(25^\circ)$
4. $\sin(155^\circ)$

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 25° and 65° are complementary (they sum to 90°), we know that cos(25°) = sin(65°). Choice A is correct because it correctly applies cos(25°) = sin(65°) using the cofunction 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

1. $55^\circ$ (correct answer)
2. $145^\circ$
3. $125^\circ$
4. $45^\circ$

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 14

1. $\sin(20^\circ) < \cos(70^\circ)$
2. $\sin(20^\circ) > \cos(70^\circ)$
3. $\sin(20^\circ)=\cos(70^\circ)$ (correct answer)
4. $\sin(20^\circ)=\sin(70^\circ)$

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 15

1. Because sine and cosine are always equal for complementary angles.
2. Because the side opposite $\theta$ is the same as the side adjacent to $90^\circ-\theta$, so both ratios use the same side over the hypotenuse. (correct answer)
3. Because $\sin^2(\theta)+\cos^2(\theta)=1$.
4. Because complementary angles add to $180^\circ$.

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. In a right triangle with acute angles θ and 90° - θ, consider a side that is opposite to angle θ (making sin(θ) = opposite/hypotenuse). That same side is adjacent to angle 90° - θ (making cos(90° - θ) = adjacent/hypotenuse = opposite/hypotenuse), showing sin(θ) = cos(90° - θ). Choice B is correct because it correctly explains that the side opposite θ is the same as the side adjacent to 90° - θ, so both ratios use the same side over the hypotenuse. Choice D incorrectly states that complementary angles add to 180°, when they actually add to 90°. 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 16

1. $\sin(A) = \cos(180^\circ - B)$
2. $\sin(A) = \sin(B)$
3. $\sin(A) = \cos(B)$ (correct answer)
4. $\sin(A) = \sec(B)$

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 C is correct because it correctly applies sin(A) = cos(B) since A and B are complementary. Choice A 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).

Question 17

1. $\cos(a + 20°) = \cos(2a - 10°)$
2. $\sin(a + 20°) = \sin(2a - 10°)$
3. $\cos(a + 20°) = \sin(2a - 10°)$ (correct answer)
4. $\tan(a + 20°) = \cot(2a - 10°)$

Explanation: When you encounter an equation where sine equals cosine, you should immediately think about complementary angle relationships. The fundamental identity $\sin \theta = \cos(90° - \theta)$ is the key to solving this type of problem. Given $\sin(a + 20°) = \cos(2a - 10°)$, you can rewrite the left side using the complementary angle identity: $\cos(90° - (a + 20°)) = \cos(2a - 10°)$, which simplifies to $\cos(70° - a) = \cos(2a - 10°)$. Since the cosines are equal, their arguments must either be equal or supplementary. For acute angle $a$, we have $70° - a = 2a - 10°$, which gives us $80° = 3a$, so $a = 26.67°$. Now, using the original complementary relationship, if $\sin(a + 20°) = \cos(2a - 10°)$, then by the cofunction identity, $\cos(a + 20°) = \sin(2a - 10°)$. This confirms answer choice C is correct. Let's examine why the other options fail: Choice A suggests $\cos(a + 20°) = \cos(2a - 10°)$, but this would mean the angles are equal or supplementary, which contradicts our sine-cosine relationship. Choice B claims $\sin(a + 20°) = \sin(2a - 10°)$, but this ignores the complementary nature of the original equation. Choice D involves tangent and cotangent, which aren't directly related to our sine-cosine equation without additional steps. Remember: when sine equals cosine, immediately think complementary angles and cofunction identities. The relationship $\sin \theta = \cos \phi$ always implies $\cos \theta = \sin \phi$.

Question 18

1. $\frac{\sqrt{9}}{4}$
2. $\frac{3}{4}$
3. $\frac{\sqrt{7}}{4}$ (correct answer)
4. $\frac{\sqrt{11}}{4}$

Explanation: When you encounter trigonometric relationships in right triangles, remember that the two acute angles are complementary (they add up to 90°), which creates a special connection between their trigonometric ratios. If one acute angle has $\cos \theta = \frac{\sqrt{7}}{4}$, you can find the sine of the other acute angle using the cofunction identity: $\sin(90° - \theta) = \cos \theta$. This means the sine of the complementary angle equals the cosine of the original angle. Therefore, $\sin(\text{other angle}) = \frac{\sqrt{7}}{4}$. You can verify this using the Pythagorean theorem. Since $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{7}}{4}$, we have adjacent = $\sqrt{7}$ and hypotenuse = 4. Using $a^2 + b^2 = c^2$: $(\sqrt{7})^2 + b^2 = 4^2$, so $7 + b^2 = 16$, giving us $b = 3$. For the other acute angle, $\sin = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{7}}{4}$. Choice A gives $\frac{\sqrt{9}}{4}$, which equals $\frac{3}{4}$ but isn't in simplest radical form. Choice B gives $\frac{3}{4}$, which would be the cosine of the other acute angle, not the sine. Choice D gives $\frac{\sqrt{11}}{4}$, which results from incorrectly adding 7 + 4 instead of using the Pythagorean theorem properly. Study tip: Remember that in right triangles, $\sin \theta = \cos(90° - \theta)$. The sine of one acute angle always equals the cosine of the other acute angle.

Question 19

1. $\cos(\theta)=\sin\left(\tfrac{\pi}{2}-\theta\right)$ (correct answer)
2. $\cos(\theta)=\cos\left(\tfrac{\pi}{2}-\theta\right)$
3. $\cos(\theta)=\sin(\theta)$
4. $\cos(\theta)=\sin(\pi-\theta)$

Explanation: This question tests understanding of the relationship between sine and cosine of complementary angles. On the unit circle, the coordinates at angle θ are (cos(θ), sin(θ)), and at the complementary angle 90° - θ, the coordinates exhibit a swap: (sin(θ), cos(θ)), demonstrating that sin(θ) = cos(90° - θ). The equation cos(θ) = sin(π/2 - θ) holds for all angles θ, meaning that if we know the cosine of any angle, we automatically know the sine of its complement. Choice A is correct because it correctly states cos(θ) = sin(π/2 - θ) from the unit circle coordinate swap. Choice B claims cos(θ) = cos(π/2 - θ), using the same function for both angles, when the cofunction relationship requires switching from cosine to sine. To verify the complementary angle relationship on the unit circle, observe that as you move from angle θ to angle π/2 - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(π/2 - 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.

Question 20

1. $\sin(35^\circ)=\sin(55^\circ)$
2. $\cos(35^\circ)=\cos(55^\circ)$
3. $\sin(35^\circ)=\cos(55^\circ)$ (correct answer)
4. $\sin(35^\circ)=\cos(145^\circ)$

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°). Choice C is correct because it correctly identifies that sin(35°) = cos(55°), applying the cofunction relationship where the sine of an angle equals the cosine of its complement. Choice A incorrectly claims sin(35°) = sin(55°), 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. 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).