In right triangle with right angle at , angle measures . If , what is the value of ?
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Geometry · Learn by Concept
Review real example questions for Sine And Cosine Of Complementary Angles in Geometry.
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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°)?
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°)?
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°).
In a right triangle, one acute angle measures (3x+15)° and has a sine value of 0.6. What is the cosine of the other acute angle in terms of the given information?
Explanation: The other acute angle measures 90° - (3x + 15)° = (75 - 3x)°. Since the angles are complementary, sin(3x + 15)° = cos(75 - 3x)° = 0.6. Choice A gives the wrong cosine value, choice B confuses the angle, and choice D uses an impossible angle measure.
If cos(2y−10)°=sin(y+25)°, what is the value of y?
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.
In right triangle DEF with right angle at E, the ratio of the side opposite angle D to the hypotenuse is 257. What is the ratio of the side adjacent to angle F to the hypotenuse?
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).
Triangle ABC has a right angle at C. If sin(A)+cos(B)=1.2, what is the value of sin(B)+cos(A)?
Explanation: When you see complementary trigonometric functions in a right triangle, remember that the acute angles are complementary, meaning they add up to 90°. This creates a special relationship between sine and cosine values. In triangle ABC with the right angle at C, angles A and B are complementary. This means A+B=90°, which gives us the key relationship: sin(A)=cos(B) and cos(A)=sin(B). Given that sin(A)+cos(B)=1.2, we can substitute using our complementary angle relationship. Since sin(A)=cos(B), we have sin(A)+sin(A)=1.2, so 2sin(A)=1.2 and sin(A)=0.6. Now we can find sin(B)+cos(A). Using our relationships again: sin(B)=cos(A) and cos(A)=sin(B). So sin(B)+cos(A)=cos(A)+sin(B)=sin(A)+cos(B)=1.2. Choice A (0.8) might tempt you if you incorrectly think the expressions should be reciprocals. Choice B (1.0) could result from assuming the Pythagorean identity applies directly here. Choice C (2.4) would come from mistakenly adding the two given expressions together. The correct answer is D (1.2). Remember: In right triangles, complementary angles create identical relationships between sine and cosine functions. When you see sin(A)+cos(B) in a right triangle, it equals cos(A)+sin(B) due to complementary angle properties.
In right triangle XYZ with right angle at Z, angle X measures a° and angle Y measures b°. If cos(a°)=k, which of the following must equal k?
Explanation: Since a° and b° are complementary angles (a + b = 90), we have cos(a°) = sin(b°) = k. Choice A represents a sum that doesn't equal k, choice B represents a product that doesn't equal k, and choice D represents sin(a°), not cos(a°).
A right triangle △JKL is shown with ∠K explicitly marked as 90∘. The acute angles are labeled θ=∠J and φ=∠L, so θ+φ=90∘. Which relationship must be true for complementary angles?
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.
In the right triangle △ABC shown, ∠C is a right angle (marked). The acute angles are labeled ∠A=θ and ∠B=φ, so θ and φ are complementary. Which statement correctly relates sin(θ) and cos(φ)?
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.
The diagram shows a right triangle △JKL with ∠K marked as a right angle. The acute angles are labeled ∠J=θ and ∠L=φ (so they are complementary). Which identity follows from the diagram?
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 at J and φ at L are complementary because their sum is 90 degrees, as the third angle is 90 degrees. For angle θ at J, the opposite side is KL, the adjacent side is JK, and the hypotenuse is JL; for angle φ at L, the opposite side is JK, the adjacent side is KL, and the hypotenuse is JL. Cosine of φ is adjacent over hypotenuse (KL/JL), while sine of θ is opposite over hypotenuse (KL/JL), showing they are equal. Therefore, cos(φ)=sin(θ), which follows from the diagram as the identity in choice A. A common distractor misconception is thinking cos(φ)=cos(θ), but complementary angles have cosines that are not equal unless both are 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.
A surveyor measures an angle of elevation of θ to the top of a building. If cos(θ)=0.8, what is sin(90°−θ)?
Explanation: This question tests your understanding of complementary angle relationships, specifically the cofunction identities. When you see an expression like sin(90°−θ), you should immediately think about how sine and cosine are related through complementary angles. The key insight is that sin(90°−θ)=cos(θ). 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, we can directly substitute to find that sin(90°−θ)=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(θ). If cos(θ)=0.8, then sin(θ)=1−0.82=0.6. However, this gives you sin(θ), not sin(90°−θ). 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°−θ)=cos(θ) and cos(90°−θ)=sin(θ). These cofunction identities appear frequently in geometry problems involving complementary angles, so memorizing them will save you time and prevent errors.