Sine and Cosine of Complementary Angles
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Pre-Calculus › Sine and Cosine of Complementary Angles
On the unit circle, the point at angle $\theta$ is $(\cos\theta,\sin\theta)$. Using the relationship between angles $\theta$ and $\tfrac{\pi}{2}-\theta$, which equation is correct?
$\cos(\theta)=\sin\left(\tfrac{\pi}{2}-\theta\right)$
$\cos(\theta)=\sin(\theta)$
$\cos(\theta)=\sin(\pi-\theta)$
$\cos(\theta)=\cos\left(\tfrac{\pi}{2}-\theta\right)$
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.
In the right triangle, one acute angle measures $35^\circ$. The other acute angle is complementary to it. What is the measure of that complementary angle?
$125^\circ$
$55^\circ$
$145^\circ$
$45^\circ$
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.
Using the complementary angle relationship, if $\sin(30^\circ)=\tfrac{1}{2}$, what is $\cos(60^\circ)$?
$\tfrac{\sqrt{3}}{2}$
$\tfrac{1}{2}$
$\tfrac{\sqrt{2}}{2}$
$1$
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.
Given the complementary angles $30^\circ$ and $60^\circ$ in a right triangle, if $\sin(30^\circ)=\tfrac{1}{2}$, what is $\cos(60^\circ)$?
$\tfrac{1}{2}$
$0$
$\tfrac{\sqrt{2}}{2}$
$\tfrac{\sqrt{3}}{2}$
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 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 C is correct because it correctly states cos(60°) = 1/2, matching sin(30°) 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.
Compare $\sin(20^\circ)$ and $\cos(70^\circ)$. Given that $20^\circ$ and $70^\circ$ are complementary, which statement is true?
$\sin(20^\circ)=\cos(70^\circ)$
$\sin(20^\circ) > \cos(70^\circ)$
$\sin(20^\circ) < \cos(70^\circ)$
$\sin(20^\circ)=\sin(70^\circ)$
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.
In a right triangle, angle $A$ is acute and angle $B$ is its complement, so $B=90^\circ-A$. Which equation correctly relates $\cos(A)$ and a sine expression involving $B$?
$\cos(A)=\cos(B)$
$\cos(A)=\csc(B)$
$\cos(A)=\sin(180^\circ-B)$
$\cos(A)=\sin(B)$
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. Since A and B are complementary (A + B = 90°), we know that cos(A) = sin(B), as cos(A) = sin(90° - A) = sin(B). Choice A is correct because it correctly applies cos(A) = sin(B) based on the complementary relationship in the right triangle. 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). 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 fact that $\sin\left(\tfrac{\pi}{6}\right)=\tfrac{1}{2}$ and that $\tfrac{\pi}{6}$ and $\tfrac{\pi}{3}$ are complementary (sum to $\tfrac{\pi}{2}$), determine $\cos\left(\tfrac{\pi}{3}\right)$.
$0$
$\tfrac{\sqrt{3}}{2}$
$\tfrac{1}{2}$
$1$
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. 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. 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 the right triangle, the acute angles are complementary. If one acute angle is $25^\circ$, which statement correctly uses the complementary angle relationship to rewrite $\sin(25^\circ)$?
$\sin(25^\circ)=\cos(155^\circ)$
$\sin(25^\circ)=\cos(65^\circ)$
$\sin(25^\circ)=\sin(65^\circ)$
$\sin(25^\circ)=\cos(25^\circ)$
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. Since 25° and 65° are complementary (they sum to 90°), we know that sin(25°) = cos(65°). Choice A is correct because it correctly applies sin(25°) = cos(65°) based on the complementary relationship in the right triangle. Choice B 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). 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 complementary angle relationship for any angle $\theta$?
$\sin(\theta)=\cos(180^\circ-\theta)$
$\sin(\theta)=\cos(90^\circ-\theta)$
$\sin(\theta)=\sin(90^\circ-\theta)$
$\sin(\theta)=\cos(\theta)$
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. 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 $\triangle ABC$ with right angle at $C$, the acute angles $A$ and $B$ are complementary (so $A+B=90^\circ$). How does $\sin(A)$ relate to $\cos(B)$?
$\sin(A)=\sec(B)$
$\sin(A)=\cos(B)$
$\sin(A)=\cos(180^\circ-B)$
$\sin(A)=\sin(B)$
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 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.