Sine and Cosine of Complementary Angles - Precalculus

In a right triangle, the acute angles $\alpha$ and $\beta$ are complementary ( $\alpha+\beta=90^\circ$ ). Which statement must be true?
Answer: $\cos(\alpha)=\sin(\beta)$ 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.
Based on the triangle shown, what is the value of $\sin \alpha \cdot \cos \beta$ ?
Answer: $\frac{9}{25}$ From the triangle, $\sin \alpha = \frac{9}{15} = \frac{3}{5}$ and $\cos \beta = \frac{9}{15} = \frac{3}{5}$ (since $\alpha$ and $\beta$ are complementary, $\cos \beta = \sin \alpha$ ). Therefore, $\sin \alpha \cdot \cos \beta = \frac{3}{5} \cdot \frac{3}{5} = \frac{9}{25}$ . Choice B uses $\sin \alpha \cdot \sin \beta$ . Choice C uses $\sin \alpha + \cos \beta$ . Choice D uses $\cos \alpha \cdot \sin \beta$ .
In a right triangle, one acute angle measures $35^\circ$ . The other acute angle is complementary to it (so the two acute angles sum to $90^\circ$ ). Using the cofunction identity, which equation is true?
Answer: $\sin(35^\circ)=\cos(55^\circ)$ 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).
In a right triangle $\triangle ABC$ with $\angle C = 90^\circ$ and $\angle A = 25^\circ$ , angles $A$ and $B$ are complementary. How does $\sin(A)$ relate to $\cos(B)$ ?
Answer: $\sin(A)=\cos(B)$ 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.
In a right triangle, one acute angle is $45^\circ$ , making the other acute angle also $45^\circ$ (they are complementary). Using the complementary angle relationship, what is $\cos(45^\circ)$ if $\sin(45^\circ)=\tfrac{\sqrt{2}}{2}$ ?
Answer: $\tfrac{\sqrt{2}}{2}$ 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 both at 45° (since 45° + 45° = 90°), we apply the relationship: sin(45°) = cos(45°) and both equal √2/2, demonstrating the cofunction property with exact values. Choice C is correct because it correctly states cos(45°) = √2/2 using the given sin(45°) = √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° and 60° (or π/6 and π/3) are complementary, so sin(30°) = cos(60°) = 1/2 and sin(60°) = cos(30°) = √3/2, but for 45°, it is its own complement.
A ladder leans against a wall, making an angle $\theta$ with the ground. If the sine of the angle the ladder makes with the wall equals $0.6$ , what is $\cos \theta$ ?
Answer: 0.6 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.
Compare $\sin(20^\circ)$ and $\cos(70^\circ)$ . Given that $20^\circ$ and $70^\circ$ are complementary, which statement is true?
Answer: $\sin(20^\circ)=\cos(70^\circ)$ 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.
Compare the two expressions $\sin(20^\circ)$ and $\cos(70^\circ)$ . Given that $20^\circ$ and $70^\circ$ are complementary angles, which statement is correct?
Answer: $\sin(20^\circ)=\cos(70^\circ)$ 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.
In a right triangle, angle $A=25^\circ$ and angle $B=65^\circ$ , so $A$ and $B$ are complementary. Which equation correctly connects $\cos(A)$ and $\sin(B)$ ?
Answer: $\cos(25^\circ)=\sin(65^\circ)$ 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.
In a right triangle $\triangle ABC$ with right angle at $C$ , the acute angles $A$ and $B$ are complementary (so $A+B=90^\circ$ ). How does $\sin(A)$ relate to $\cos(B)$ ?
Answer: $\sin(A)=\cos(B)$ 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 B is correct because it correctly applies sin(A) = cos(B) based on the complementary relationship in the right triangle. Choice A claims sin(A) = sin(B), using the same function for both angles, when the cofunction relationship requires switching from sine to 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.
Using the fact that $\sin\left(\tfrac{\pi}{6}\right)=\tfrac{1}{2}$ and that $\tfrac{\pi}{6}$ and $\tfrac{\pi}{3}$ are complementary (they sum to $\tfrac{\pi}{2}$ ), determine $\cos\left(\tfrac{\pi}{3}\right)$ .
Answer: $\tfrac{1}{2}$ 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, matching sin(π/6) via 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. 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.
If $\cos(2x - 15°) = \sin(3x + 10°)$ and both angles are acute, what is the value of $x$ ?
Answer: 19° When you encounter an equation where cosine equals sine, remember the fundamental cofunction identity: $\cos \theta = \sin(90° - \theta)$ . This means cosine of an angle equals sine of its complement. Using this identity, we can rewrite $\cos(2x - 15°) = \sin(3x + 10°)$ as: $\sin(90° - (2x - 15°)) = \sin(3x + 10°)$ Simplifying the left side: $\sin(90° - 2x + 15°) = \sin(3x + 10°)$ $\sin(105° - 2x) = \sin(3x + 10°)$ When two sine functions are equal, their angles must be equal (assuming we're in the principal domain): $105° - 2x = 3x + 10°$ Solving for $x$ : $105° - 10° = 3x + 2x$ $95° = 5x$ $x = 19°$ Let's verify: $\cos(2(19°) - 15°) = \cos(23°)$ and $\sin(3(19°) + 10°) = \sin(67°)$ . Since $23° + 67° = 90°$ , these are indeed equal. Choice (A) 25° gives us $\cos(35°)$ and $\sin(85°)$ , but $35° + 85° = 120° \neq 90°$ . Choice (B) 21° yields $\cos(27°)$ and $\sin(73°)$ , where $27° + 73° = 100°$ . Choice (C) 23° produces $\cos(31°)$ and $\sin(79°)$ , where $31° + 79° = 110°$ . Strategy tip: When cosine equals sine, immediately think cofunction identity. The two angles inside must be complementary (sum to 90°). This transforms a trigonometric equation into a simple algebraic one.
In a right triangle, one acute angle is $25^\circ$ and the other is $65^\circ$ , so the angles are complementary. If $\cos(25^\circ)$ is known, which expression is equal to it by the cofunction relationship?
Answer: $\sin(65^\circ)$ 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.
Given that $30^\circ$ and $60^\circ$ are complementary angles (they sum to $90^\circ$ ) and $\sin(30^\circ)=\tfrac{1}{2}$ , what is $\cos(60^\circ)$ ?
Answer: $\tfrac{1}{2}$ 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. 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 B is correct because it correctly applies sin(30°) = cos(60°) = 1/2 using the complementary angle relationship. Choice A incorrectly gives √3/2, which is actually cos(30°) or sin(60°), confusing which special angle value corresponds to which function. 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.
Using the fact that $\sin\left(\frac{\pi}{6}\right)=\tfrac{1}{2}$ and that $\frac{\pi}{6}$ and $\frac{\pi}{3}$ are complementary (sum to $\frac{\pi}{2}$ ), determine $\cos\left(\frac{\pi}{3}\right)$ .
Answer: $\tfrac{1}{2}$ 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.
Given the complementary angles $30^\circ$ and $60^\circ$ (they sum to $90^\circ$ ) and the fact that $\sin(30^\circ)=\tfrac{1}{2}$ , what is $\cos(60^\circ)$ ?
Answer: $\tfrac{1}{2}$ 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 30° and 60°, we apply the relationship: sin(30°) = cos(60°) and both equal 1/2, while sin(60°) = cos(30°) and both equal √3/2, demonstrating the cofunction property with exact values. Choice B is correct because it correctly states cos(60°) = 1/2 using the given sin(30°) = 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. 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.
Using the cofunction identity for complementary angles, which equation correctly expresses the relationship between sine and cosine?
Answer: $\sin(\theta)=\cos(90^\circ-\theta)$ 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.
Which equation correctly expresses the complementary angle relationship for degrees?
Answer: $\sin(\theta)=\cos(90^\circ-\theta)$ 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.
Using the complementary angle relationship on the unit circle, if $\theta$ is an acute angle, what is the relationship between $\cos(\theta)$ and $\sin(90^\circ-\theta)$ ?
Answer: $\cos(\theta)=\sin(90^\circ-\theta)$ 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 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 that cos(θ) = sin(90° - θ), which is the other form of the complementary angle relationship equivalent to sin(θ) = cos(90° - θ). Choice B incorrectly claims cos(θ) = cos(90° - 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 90° - θ, the x and y coordinates swap positions, showing cos(θ) ↔ sin(90° - θ).
Angles $20^\circ$ and $70^\circ$ are complementary. Compare $\sin(20^\circ)$ and $\cos(70^\circ)$ .
Answer: $\sin(20^\circ)=\cos(70^\circ)$ 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 identifies that sin(20°) = cos(70°) using the complementary angle relationship. Choice C incorrectly suggests sin(20°) < cos(70°), failing to recognize that these values are equal due to the complementary angle relationship. 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.

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