Geometry Quiz: Sine And Cosine Of Complementary Angles
20 questions · exam conditions
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Sine And Cosine Of Complementary AnglesQuestion 1 of 20

Triangle PQRPQR is a right triangle with the right angle at QQ. If cos⁡(∠P)=513\cos(\angle P) = \frac{5}{13}, what is sin⁡(∠R)\sin(\angle R)?

1213\frac{12}{13}
513\frac{5}{13}
135\frac{13}{5}
512\frac{5}{12}
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Geometry Quiz

Geometry Quiz: Sine And Cosine Of Complementary Angles

Practice Sine And Cosine Of Complementary Angles in Geometry 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 Geometry.

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.

All questions

Question 1

Triangle PQRPQR is a right triangle with the right angle at QQ. If cos⁡(∠P)=513\cos(\angle P) = \frac{5}{13}, what is sin⁡(∠R)\sin(\angle R)?

  1. 1213\frac{12}{13}
  2. 513\frac{5}{13} (correct answer)
  3. 135\frac{13}{5}
  4. 512\frac{5}{12}
Explanation: In right triangle PQR, angles P and R are complementary (sum to 90°). Therefore, cos(∠P) = sin(∠R) = 5/13. Choice A represents sin(∠P), choice C represents sec(∠P), and choice D represents tan(∠R).

Question 2

In right triangle ABCABC with right angle at CC, angle AA measures 32°32°. If sin⁡(32°)=0.53\sin(32°) = 0.53, what is the value of cos⁡(58°)\cos(58°)?

  1. 0.530.53 (correct answer)
  2. 0.850.85
  3. 0.470.47
  4. 1.881.88
Explanation: Since angles A and B are complementary in a right triangle, angle B = 90° - 32° = 58°. By the complementary angle relationship, sin(32°) = cos(58°) = 0.53. Choice B represents cos(32°), choice C represents sin(58°), and choice D represents sec(32°).

Question 3

If cos⁡(2y−10)°=sin⁡(y+25)°\cos(2y - 10)° = \sin(y + 25)°, what is the value of yy?

  1. 2525 (correct answer)
  2. 3535
  3. 1515
  4. 4545
Explanation: For cos(A) = sin(B), the angles must be complementary: A + B = 90°. So (2y - 10) + (y + 25) = 90, which gives 3y + 15 = 90, therefore 3y = 75 and y = 25. The other choices result from common algebraic errors or incorrect complementary angle relationships.

Question 4

In right triangle DEFDEF with right angle at EE, the ratio of the side opposite angle DD to the hypotenuse is 725\frac{7}{25}. What is the ratio of the side adjacent to angle FF to the hypotenuse?

  1. 2425\frac{24}{25}
  2. 725\frac{7}{25} (correct answer)
  3. 257\frac{25}{7}
  4. 724\frac{7}{24}
Explanation: The side opposite angle D is the same as the side adjacent to angle F (both refer to side EF). Since sin(D) = 7/25, and angles D and F are complementary, cos(F) = sin(D) = 7/25. Choice A represents cos(D), choice C represents csc(D), and choice D represents tan(D).

Question 5

A right triangle △JKL\triangle JKL is shown with ∠K\angle K explicitly marked as 90∘90^\circ. The acute angles are labeled θ=∠J\theta=\angle J and φ=∠L\varphi=\angle L, so θ+φ=90∘\theta+\varphi=90^\circ. Which relationship must be true for complementary angles?

  1. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  2. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=1\sin(\theta)=1
Explanation: This problem tests understanding of the sine-cosine relationship for complementary angles. Since angle K is marked as 90° and θ = angle J and φ = angle L are the acute angles with θ + φ = 90°, these angles are complementary. For angle θ at vertex J, the opposite side is KL and the adjacent side is JK, giving sin(θ) = KL/JL. For angle φ at vertex L, the opposite side is JK and the adjacent side is KL, giving cos(φ) = KL/JL. Because both ratios equal KL/JL, we conclude sin(θ) = cos(φ). A common error is thinking sin(θ) = sin(φ), but complementary angles don't have equal sines unless they're both 45°. To master this concept, always identify which side is opposite and which is adjacent to each angle before applying trigonometric definitions.

Question 6

In the right triangle △ABC\triangle ABC shown, ∠C\angle C is a right angle (marked). The acute angles are labeled ∠A=θ\angle A = \theta and ∠B=φ\angle B = \varphi, so θ\theta and φ\varphi are complementary. Which statement correctly relates sin⁡(θ)\sin(\theta) and cos⁡(φ)\cos(\varphi)?

  1. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  2. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. In right triangle ABC with right angle at C, angles θ at A and φ at B are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle θ at A, the opposite side is BC, the adjacent side is AC, and the hypotenuse is AB; for angle φ at B, the opposite side is AC, the adjacent side is BC, and the hypotenuse is AB. Sine of θ is opposite over hypotenuse (BC/AB), while cosine of φ is adjacent over hypotenuse (BC/AB), showing they are equal. Therefore, sin(θ) = cos(φ), which correctly relates them as in choice B. A common distractor misconception is assuming sin(θ) = sin(φ), but since θ and φ are different angles, their sines are generally not equal unless θ = φ = 45°. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 7

The diagram shows a right triangle △JKL\triangle JKL with ∠K\angle K marked as a right angle. The acute angles are labeled ∠J=θ\angle J=\theta and ∠L=φ\angle L=\varphi (so they are complementary). Which identity follows from the diagram?

  1. cos⁡(φ)=sin⁡(θ)\cos(\varphi)=\sin(\theta) (correct answer)
  2. cos⁡(φ)=cos⁡(θ)\cos(\varphi)=\cos(\theta)
  3. cos⁡(φ)=tan⁡(θ)\cos(\varphi)=\tan(\theta)
  4. cos⁡(φ)=sin⁡(90∘)\cos(\varphi)=\sin(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. In right triangle JKL with right angle at K, angles θ\theta at J and φ\varphi at L are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle θ\theta at J, the opposite side is KL, the adjacent side is JK, and the hypotenuse is JL; for angle φ\varphi at L, the opposite side is JK, the adjacent side is KL, and the hypotenuse is JL. Cosine of φ\varphi is adjacent over hypotenuse (KL/JL), while sine of θ\theta is opposite over hypotenuse (KL/JL), showing they are equal. Therefore, cos⁡(φ)=sin⁡(θ)\cos(\varphi) = \sin(\theta), which follows from the diagram as the identity in choice A. A common distractor misconception is thinking cos⁡(φ)=cos⁡(θ)\cos(\varphi) = \cos(\theta), but complementary angles have cosines that are not equal unless both are 45∘45^\circ. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 8

A surveyor measures an angle of elevation of θ\theta to the top of a building. If cos⁡(θ)=0.8\cos(\theta) = 0.8, what is sin⁡(90°−θ)\sin(90° - \theta)?

  1. 0.60.6
  2. 0.750.75
  3. 1.251.25
  4. 0.80.8 (correct answer)
Explanation: This question tests your understanding of complementary angle relationships, specifically the cofunction identities. When you see an expression like sin⁡(90°−θ)\sin(90° - \theta), you should immediately think about how sine and cosine are related through complementary angles. The key insight is that sin⁡(90°−θ)=cos⁡(θ)\sin(90° - \theta) = \cos(\theta). This is one of the fundamental cofunction identities: the sine of an angle equals the cosine of its complement. Since we're given that cos⁡(θ)=0.8\cos(\theta) = 0.8, we can directly substitute to find that sin⁡(90°−θ)=0.8\sin(90° - \theta) = 0.8. Let's examine why the other answers are incorrect. Choice A (0.60.6) likely comes from using the Pythagorean identity to find sin⁡(θ)\sin(\theta). If cos⁡(θ)=0.8\cos(\theta) = 0.8, then sin⁡(θ)=1−0.82=0.6\sin(\theta) = \sqrt{1 - 0.8^2} = 0.6. However, this gives you sin⁡(θ)\sin(\theta), not sin⁡(90°−θ)\sin(90° - \theta). Choice B (0.750.75) doesn't correspond to any standard trigonometric calculation with the given information and may represent a computational error. Choice C (1.251.25) is impossible since sine values must be between -1 and 1, making this a clear distractor for students who might make algebraic mistakes. Remember this pattern: sin⁡(90°−θ)=cos⁡(θ)\sin(90° - \theta) = \cos(\theta) and cos⁡(90°−θ)=sin⁡(θ)\cos(90° - \theta) = \sin(\theta). These cofunction identities appear frequently in geometry problems involving complementary angles, so memorizing them will save you time and prevent errors.

Question 9

In the shown right triangle △MNO\triangle MNO, ∠N\angle N is marked as a right angle, with acute angles θ=∠M\theta=\angle M and ϕ=∠O\phi=\angle O. Which statement correctly relates cos⁡(θ)\cos(\theta) and sin⁡(ϕ)\sin(\phi)?

  1. cos⁡(θ)=sin⁡(ϕ)\cos(\theta)=\sin(\phi) (correct answer)
  2. cos⁡(θ)=cos⁡(ϕ)\cos(\theta)=\cos(\phi)
  3. cos⁡(θ)=sin⁡(θ)\cos(\theta)=\sin(\theta)
  4. cos⁡(θ)=0\cos(\theta)=0
Explanation: This problem focuses on the cosine-sine relationship for complementary angles. In triangle MNO with right angle at N, angles θ = ∠M and φ = ∠O are complementary because θ + φ = 90°. For angle θ, the opposite side is NO and the adjacent side is MN, giving cos(θ) = MN/MO. For angle φ, the opposite side is MN and the adjacent side is NO, giving sin(φ) = MN/MO. Since both equal MN/MO, we have cos(θ) = sin(φ). The distractor cos(θ) = cos(φ) wrongly assumes complementary angles have equal cosine values. Remember that in a right triangle, each acute angle's adjacent side is the other acute angle's opposite side.

Question 10

In right triangle △GHI\triangle GHI, ∠H\angle H is marked as 90∘90^\circ. The acute angles are labeled ∠G=θ\angle G=\theta and ∠I=φ\angle I=\varphi, so they are complementary. Which statement correctly relates cos⁡(θ)\cos(\theta) and sin⁡(φ)\sin(\varphi)?

  1. cos⁡(θ)=sin⁡(φ)\cos(\theta)=\sin(\varphi) (correct answer)
  2. cos⁡(θ)=cos⁡(φ)\cos(\theta)=\cos(\varphi)
  3. cos⁡(θ)=tan⁡(φ)\cos(\theta)=\tan(\varphi)
  4. cos⁡(θ)=sin⁡(90∘)\cos(\theta)=\sin(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. In right triangle GHI with right angle at H, angles θ at G and φ at I are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle θ at G, the opposite side is HI, the adjacent side is GH, and the hypotenuse is GI; for angle φ at I, the opposite side is GH, the adjacent side is HI, and the hypotenuse is GI. Cosine of θ is adjacent over hypotenuse (GH/GI), while sine of φ is opposite over hypotenuse (GH/GI), showing they are equal. Therefore, cos(θ) = sin(φ), which correctly relates them as in choice A. A common distractor misconception is confusing it with cos(θ) = cos(φ), but cosines of complementary angles are not equal. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 11

In the diagram, △MNO\triangle MNO is a right triangle with the right angle at NN (marked). The acute angles are labeled θ\theta at MM and φ\varphi at OO, making them complementary. Which identity follows from the diagram?

  1. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  2. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=sin⁡(90∘)\sin(\theta)=\sin(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. The acute angles θ and φ are complementary because their sum is 90 degrees, as they are the two non-right angles in the right triangle at N. For angle θ at M, the opposite side is NO, the adjacent side is MN, and the hypotenuse is MO; for angle φ at O, the opposite side is MN, the adjacent side is NO, and the hypotenuse is MO. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, so sin(θ) = NO/MO and cos(φ) = NO/MO. Therefore, sin(θ) equals cos(φ) because they both represent the same ratio of sides. A common misconception is thinking sin(θ) = sin(φ), which would require equal opposite sides relative to the hypotenuse, but they swap. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 12

A right triangle △DEF\triangle DEF is shown with ∠E\angle E marked as 90∘90^\circ. The acute angles are labeled θ\theta at DD and ϕ\phi at FF, so θ\theta and ϕ\phi are complementary. Which relationship must be true for complementary angles?

  1. sin⁡(θ)=sin⁡(ϕ)\sin(\theta)=\sin(\phi)
  2. cos⁡(θ)=cos⁡(ϕ)\cos(\theta)=\cos(\phi)
  3. sin⁡(θ)=cos⁡(ϕ)\sin(\theta)=\cos(\phi) (correct answer)
  4. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
Explanation: The skill here is understanding the relationship between sine and cosine for complementary angles in a right triangle. In a right triangle, the two acute angles θ and φ add up to 90 degrees because the third angle is 90 degrees, making them complementary. For angle θ at D, the opposite side is EF and the adjacent side is DE; for φ at F, the opposite side is DE and the adjacent side is EF. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, linking the trigonometric functions of the two angles. Therefore, sin(θ) equals cos(φ) since they share the same ratio. A common distractor misconception is equating sin(θ) to cos(θ), which ignores the complementary nature and side roles. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides for each angle to see how their roles swap.

Question 13

In the right triangle △ABC\triangle ABC shown, ∠C\angle C is a right angle, and the acute angles are labeled θ=∠A\theta=\angle A and φ=∠B\varphi=\angle B (so θ\theta and φ\varphi are complementary). Which statement correctly relates sin⁡(θ)\sin(\theta) and cos⁡(φ)\cos(\varphi)?

  1. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  2. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  3. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
  4. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
Explanation: This question tests the sine-cosine relationship for complementary angles in a right triangle. Since angle C is the right angle and θ = angle A and φ = angle B are the acute angles, we know that θ + φ = 90°, making them complementary. For angle θ at vertex A, the opposite side is BC and the adjacent side is AC, so sin(θ) = BC/AB. For angle φ at vertex B, the opposite side is AC and the adjacent side is BC, so cos(φ) = BC/AB. Since both expressions equal BC/AB, we have sin(θ) = cos(φ). A common misconception is thinking that complementary angles have equal sines, but the key insight is that the same side plays different roles (opposite vs adjacent) for the two angles. To verify this relationship, redraw the triangle and label which sides are opposite and adjacent to each angle.

Question 14

In right triangle △XYZ\triangle XYZ, ∠Y\angle Y is marked as 90∘90^\circ. The acute angles are labeled ∠X=θ\angle X=\theta and ∠Z=φ\angle Z=\varphi (complementary). Which relationship must be true for complementary angles in this right triangle?

  1. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  2. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. In right triangle XYZ with right angle at Y, angles θ\theta at X and φ\varphi at Z are complementary because their sum is 90∘90^\circ, as the third angle is 90∘90^\circ. For angle θ\theta at X, the opposite side is YZ, the adjacent side is XY, and the hypotenuse is XZ; for angle φ\varphi at Z, the opposite side is XY, the adjacent side is YZ, and the hypotenuse is XZ. Sine of θ\theta is opposite over hypotenuse (YZXZ\frac{\text{YZ}}{\text{XZ}}), while cosine of φ\varphi is adjacent over hypotenuse (YZXZ\frac{\text{YZ}}{\text{XZ}}), showing they are equal. Therefore, sin⁡(θ)=cos⁡(φ)\sin(\theta) = \cos(\varphi), which must be true for these complementary angles as in choice A. A common distractor misconception is assuming sin⁡(θ)=cos⁡(θ)\sin(\theta) = \cos(\theta), but sine and cosine are equal only for 45∘45^\circ angles, not generally. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 15

In the shown right triangle, ∠B\angle B is marked as 90∘90^\circ. The acute angles are labeled θ=∠A\theta=\angle A and φ=∠C\varphi=\angle C, so θ+φ=90∘\theta+\varphi=90^\circ. Which statement correctly relates cos⁡(θ)\cos(\theta) and sin⁡(φ)\sin(\varphi)?

  1. cos⁡(θ)=cos⁡(φ)\cos(\theta)=\cos(\varphi)
  2. cos⁡(θ)=sin⁡(φ)\cos(\theta)=\sin(\varphi) (correct answer)
  3. cos⁡(θ)=sin⁡(θ)\cos(\theta)=\sin(\theta)
  4. cos⁡(θ)=sin⁡(180∘−φ)\cos(\theta)=\sin(180^\circ-\varphi)
Explanation: This question focuses on the cosine-sine relationship for complementary angles. With angle B marked as 90° and θ = angle A and φ = angle C as the acute angles where θ + φ = 90°, these angles are complementary. For angle θ at vertex A, the opposite side is BC and the adjacent side is AB, giving cos(θ) = AB/AC. For angle φ at vertex C, the opposite side is AB and the adjacent side is BC, giving sin(φ) = AB/AC. Since both expressions equal AB/AC, we conclude cos(θ) = sin(φ). The incorrect option cos(θ) = cos(φ) suggests complementary angles have equal cosines, but this misses the key insight that the same side serves different roles for each angle. To master this concept, always identify which side is opposite and which is adjacent to each angle before applying definitions.

Question 16

In the right triangle △MNO\triangle MNO shown, ∠N\angle N is explicitly marked as 90∘90^\circ. The acute angles are labeled ∠M=θ\angle M=\theta and ∠O=φ\angle O=\varphi, so θ\theta and φ\varphi are complementary. Which statement correctly relates sin⁡(θ)\sin(\theta) and cos⁡(φ)\cos(\varphi)?

  1. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  2. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  3. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. In right triangle MNO with right angle at N, angles θ at M and φ at O are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle θ at M, the opposite side is NO, the adjacent side is MN, and the hypotenuse is MO; for angle φ at O, the opposite side is MN, the adjacent side is NO, and the hypotenuse is MO. Sine of θ is opposite over hypotenuse (NO/MO), while cosine of φ is adjacent over hypotenuse (NO/MO), showing they are equal. Therefore, sin(θ) = cos(φ), which correctly relates them as in choice A. A common distractor misconception is assuming sin(θ) = sin(φ), but this holds only if θ = φ, which is not general for complementary angles. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.

Question 17

In the diagram, △ABC\triangle ABC is a right triangle drawn in the plane with a right angle at CC. The acute angles are labeled θ\theta at AA and ϕ\phi at BB, so θ\theta and ϕ\phi are complementary. Which statement correctly relates sin⁡(θ)\sin(\theta) and cos⁡(ϕ)\cos(\phi)?

  1. sin⁡(θ)=sin⁡(ϕ)\sin(\theta)=\sin(\phi)
  2. sin⁡(θ)=cos⁡(ϕ)\sin(\theta)=\cos(\phi) (correct answer)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
Explanation: The skill here is understanding the relationship between sine and cosine for complementary angles in a right triangle. In a right triangle, the two acute angles θ and φ add up to 90 degrees because the third angle is 90 degrees, making them complementary. For angle θ at A, the opposite side is BC and the adjacent side is AC; for φ at B, the opposite side is AC and the adjacent side is BC. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, so these definitions show that the opposite side to θ is the adjacent side to φ, and vice versa. Therefore, sin(θ) equals cos(φ) because both are the same ratio of the shared side over the hypotenuse. A common distractor misconception is believing sin(θ) = sin(φ), but this only holds if θ equals φ, which is not generally true for complementary angles. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides for each angle to see how their roles swap.

Question 18

In the diagram, △GHI\triangle GHI is a right triangle with the right angle explicitly marked at HH. The acute angles are labeled θ\theta at GG and ϕ\phi at II, so ϕ=90∘−θ\phi=90^\circ-\theta. Which expression represents cos⁡(θ)\cos(\theta) in terms of sin⁡(ϕ)\sin(\phi)?

  1. cos⁡(θ)=sin⁡(ϕ)\cos(\theta)=\sin(\phi) (correct answer)
  2. cos⁡(θ)=cos⁡(ϕ)\cos(\theta)=\cos(\phi)
  3. cos⁡(θ)=sin⁡(θ)\cos(\theta)=\sin(\theta)
  4. cos⁡(θ)=sin⁡(90∘)\cos(\theta)=\sin(90^\circ)
Explanation: The skill here is understanding the relationship between sine and cosine for complementary angles in a right triangle. In a right triangle, the two acute angles θ and φ add up to 90 degrees because the third angle is 90 degrees, making them complementary. For angle θ at G, the opposite side is HI and the adjacent side is GH; for φ at I, the opposite side is GH and the adjacent side is HI. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, connecting sin(φ) to cos(θ). Therefore, cos(θ) equals sin(φ) based on the swapped side roles. A common distractor misconception is equating cos(θ) to cos(φ), which fails to account for the complementary difference. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides for each angle to see how their roles swap.

Question 19

In the diagram, △JKL\triangle JKL is a right triangle with the right angle explicitly marked at KK. The acute angles are labeled θ\theta at JJ and ϕ\phi at LL, so θ+ϕ=90∘\theta+\phi=90^\circ. Which identity follows from the diagram?

  1. cos⁡(ϕ)=sin⁡(θ)\cos(\phi)=\sin(\theta) (correct answer)
  2. cos⁡(ϕ)=cos⁡(θ)\cos(\phi)=\cos(\theta)
  3. cos⁡(ϕ)=sin⁡(90∘)\cos(\phi)=\sin(90^\circ)
  4. cos⁡(ϕ)=cos⁡(ϕ/2)\cos(\phi)=\cos(\phi/2)
Explanation: The skill here is understanding the relationship between sine and cosine for complementary angles in a right triangle. In a right triangle, the two acute angles θ and φ add up to 90 degrees because the third angle is 90 degrees, making them complementary. For angle θ at J, the opposite side is KL and the adjacent side is JK; for φ at L, the opposite side is JK and the adjacent side is KL. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, so the definitions interchange the side roles. Therefore, cos(φ) equals sin(θ) because both are the same fractional part of the hypotenuse. A common distractor misconception is thinking cos(φ) = cos(θ), but this would require the angles to be identical, not complementary. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides for each angle to see how their roles swap.

Question 20

A right triangle △RST\triangle RST is drawn with the right angle at SS (marked). The acute angles are labeled θ\theta at RR and φ\varphi at TT. Which relationship depends on the angles being complementary?

  1. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  2. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)
Explanation: The skill here is understanding the sine-cosine relationship for complementary angles in a right triangle. The acute angles θ and φ are complementary because their sum is 90 degrees, as they are the two non-right angles in the right triangle at S. For angle θ at R, the opposite side is ST, the adjacent side is RS, and the hypotenuse is RT; for angle φ at T, the opposite side is RS, the adjacent side is ST, and the hypotenuse is RT. Sine is defined as opposite over hypotenuse, and cosine as adjacent over hypotenuse, so sin(θ) = ST/RT and cos(φ) = ST/RT. Therefore, sin(θ) equals cos(φ) because they both represent the same ratio of sides. A common misconception is thinking sin(θ) = cos(θ), but that mixes up the side definitions. To transfer this strategy, redraw the triangle and label the opposite and adjacent sides relative to each angle to see how they swap roles.