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Geometry Help: Sine And Cosine Of Complementary Angles

Review real example questions for Sine And Cosine Of Complementary Angles in Geometry.

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Triangle $PQR$ is a right triangle with the right angle at $Q$ . If $\cos(\angle P) = \frac{5}{13}$ , what is $\sin(\angle R)$ ?

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Question 1

Triangle $PQR$ is a right triangle with the right angle at $Q$ . If $\cos(\angle P) = \frac{5}{13}$ , what is $\sin(\angle R)$ ?

$\frac{12}{13}$
$\frac{5}{13}$ (correct answer)
$\frac{13}{5}$
$\frac{5}{12}$

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 $ABC$ with right angle at $C$ , angle $A$ measures $32°$ . If $\sin(32°) = 0.53$ , what is the value of $\cos(58°)$ ?

$0.53$ (correct answer)
$0.85$
$0.47$
$1.88$

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

$25$ (correct answer)
$35$
$15$
$45$

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 $DEF$ with right angle at $E$ , the ratio of the side opposite angle $D$ to the hypotenuse is $\frac{7}{25}$ . What is the ratio of the side adjacent to angle $F$ to the hypotenuse?

$\frac{24}{25}$
$\frac{7}{25}$ (correct answer)
$\frac{25}{7}$
$\frac{7}{24}$

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

$\sin(\theta)=\sin(\varphi)$
$\sin(\theta)=\cos(\varphi)$ (correct answer)
$\sin(\theta)=\cos(\theta)$
$\sin(\theta)=1$

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

$\sin(\theta)=\sin(\varphi)$
$\sin(\theta)=\cos(\varphi)$ (correct answer)
$\sin(\theta)=\cos(\theta)$
$\sin(\theta)=\cos(90^\circ)$

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

$\cos(\varphi)=\sin(\theta)$ (correct answer)
$\cos(\varphi)=\cos(\theta)$
$\cos(\varphi)=\tan(\theta)$
$\cos(\varphi)=\sin(90^\circ)$

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 $\theta$ at J and $\varphi$ at L are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle $\theta$ at J, the opposite side is KL, the adjacent side is JK, and the hypotenuse is JL; for angle $\varphi$ at L, the opposite side is JK, the adjacent side is KL, and the hypotenuse is JL. Cosine of $\varphi$ is adjacent over hypotenuse (KL/JL), while sine of $\theta$ is opposite over hypotenuse (KL/JL), showing they are equal. Therefore, $\cos(\varphi) = \sin(\theta)$ , which follows from the diagram as the identity in choice A. A common distractor misconception is thinking $\cos(\varphi) = \cos(\theta)$ , but complementary angles have cosines that are not equal unless both are $45^\circ$ . 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 $\theta$ to the top of a building. If $\cos(\theta) = 0.8$ , what is $\sin(90° - \theta)$ ?

$0.6$
$0.75$
$1.25$
$0.8$ (correct answer)

Explanation: This question tests your understanding of complementary angle relationships, specifically the cofunction identities. When you see an expression like $\sin(90° - \theta)$ , you should immediately think about how sine and cosine are related through complementary angles. The key insight is that $\sin(90° - \theta) = \cos(\theta)$ . This is one of the fundamental cofunction identities: the sine of an angle equals the cosine of its complement. Since we're given that $\cos(\theta) = 0.8$ , we can directly substitute to find that $\sin(90° - \theta) = 0.8$ . Let's examine why the other answers are incorrect. Choice A ( $0.6$ ) likely comes from using the Pythagorean identity to find $\sin(\theta)$ . If $\cos(\theta) = 0.8$ , then $\sin(\theta) = \sqrt{1 - 0.8^2} = 0.6$ . However, this gives you $\sin(\theta)$ , not $\sin(90° - \theta)$ . Choice B ( $0.75$ ) doesn't correspond to any standard trigonometric calculation with the given information and may represent a computational error. Choice C ( $1.25$ ) 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: $\sin(90° - \theta) = \cos(\theta)$ and $\cos(90° - \theta) = \sin(\theta)$ . 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 $\triangle MNO$ , $\angle N$ is marked as a right angle, with acute angles $\theta=\angle M$ and $\phi=\angle O$ . Which statement correctly relates $\cos(\theta)$ and $\sin(\phi)$ ?

$\cos(\theta)=\sin(\phi)$ (correct answer)
$\cos(\theta)=\cos(\phi)$
$\cos(\theta)=\sin(\theta)$
$\cos(\theta)=0$

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

$\cos(\theta)=\sin(\varphi)$ (correct answer)
$\cos(\theta)=\cos(\varphi)$
$\cos(\theta)=\tan(\varphi)$
$\cos(\theta)=\sin(90^\circ)$

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.