Trigonometry - ACT Math

1. In a right triangle, if the opposite side to angle $\theta$ is 5 and the adjacent side is 12, what is $\tan(\theta)$?
   Answer: 5/12 For angle $\theta$, the opposite side is 5 and the adjacent side is 12. Using SOH-CAH-TOA, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Therefore, $\tan(\theta) = \frac{5}{12}$. Choice B ($\frac{12}{5}$) incorrectly flipped the ratio, using adjacent/opposite instead.
2. In the right triangle shown, the right angle is at $A$. The side lengths are $AB=20$, $AC=21$, and $BC=29$. Angle $\theta$ is at $B$. What is $\cos(\theta)$?
   Answer: $\frac{20}{29}$ For angle θ at vertex B, we identify the sides relative to this angle. The adjacent side to angle θ is AB = 20, the opposite side is AC = 21, and the hypotenuse is BC = 29. Using SOH-CAH-TOA, cos(θ) = adjacent/hypotenuse = AB/BC = 20/29. Choice A gives 21/29, which would be using the opposite side instead of the adjacent side.
3. Which equals $\sin(0^\circ)$?
   Answer: $0$ The angle 0° represents a position on the positive x-axis of the unit circle. At this position, the y-coordinate is 0, so sin(0°) = 0. This is a fundamental trigonometric value where the angle has no vertical component. Choice A gives 1, which is actually cos(0°), not sin(0°).
4. In the right triangle shown, if the opposite side to angle $\theta$ is 7 and the hypotenuse is 25, what is $\sin(\theta)$?
   Answer: 7/25 For angle θ, the opposite side is 7 and the hypotenuse is 25. Using SOH-CAH-TOA, sin(θ) = opposite/hypotenuse. Therefore, sin(θ) = 7/25. Choice B (24/25) would be the cosine if the adjacent side were 24.
5. In the right triangle, if the adjacent side is 9 and the hypotenuse is 15, what is $\cos(\theta)$?
   Answer: $0.6$ For angle θ, the adjacent side is 9 and the hypotenuse is 15. Using SOH-CAH-TOA, cos(θ) = adjacent/hypotenuse. Therefore, cos(θ) = 9/15 = 3/5 = 0.6. Choice B (0.8) would be 12/15, which might confuse opposite over hypotenuse (sine), not cosine.
6. What is $\sin(\theta)$ for the right triangle shown, where the opposite side is 3 and the hypotenuse is 5?
   Answer: 0.6 For angle θ, the opposite side is 3 and the hypotenuse is 5. Using SOH-CAH-TOA, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Therefore, $\sin(\theta) = \frac{3}{5} = 0.6$. Choice B (0.8) would be $\frac{4}{5}$, which might confuse adjacent over hypotenuse ($\cos(\theta)$), not sine.
7. What is $\sin(30^\circ)$?​​
   Answer: $\frac{1}{2}$ The angle 30° is a special angle on the unit circle. sin(30°) = 1/2, which is a standard value to memorize. Choice A shows √3/2, which is actually cos(30°), not sin(30°).
8. In the right triangle shown, if the adjacent side to angle $\theta$ is 4 and the hypotenuse is 5, what is $\cos(\theta)$?
   Answer: 4/5 For angle θ, the adjacent side is 4 and the hypotenuse is 5. Using SOH-CAH-TOA, cos(θ) = adjacent/hypotenuse. Therefore, cos(θ) = 4/5. Choice B (3/5) would be sin(θ) if the opposite side were 3.
9. What angle has $\sin(\theta) = \frac{\sqrt{3}}{2}$?
   Answer: $60^\circ$ We need to find the angle where sin(θ) = √3/2. Using unit circle values, sin(60°) = √3/2. This is a standard trigonometric value. Choice A (30°) has sin(30°) = 1/2, not √3/2.
10. A right triangle has legs of lengths $5$ (adjacent to angle $\theta$) and $12$ (opposite to angle $\theta$). What is $\cos(\theta)$?​​
    Answer: $\frac{5}{13}$ For angle θ, adjacent = 5 and opposite = 12, so the hypotenuse = √(5² + 12²) = √169 = 13. Using CAH: cos = adjacent/hypotenuse, we get cos(θ) = 5/13. Choice A shows 12/13, which would be sin(θ) using the opposite side.
11. In a right triangle, if the opposite side to angle $\theta$ is 15 and the adjacent side is 20, what is $\tan(\theta)$?
    Answer: 3/4 For angle θ, the opposite side is 15 and the adjacent side is 20. Using SOH-CAH-TOA, tan(θ) = opposite/adjacent = 15/20. This simplifies to 3/4. Choice B (4/3) incorrectly flipped the ratio, using adjacent/opposite instead.
12. In a right triangle, the angle $\theta$ has opposite side length $3$ and adjacent side length $4$. What is $\tan(\theta)$?​​
    Answer: $\frac{3}{4}$ For angle θ, the opposite side = 3 and the adjacent side = 4. Using TOA: tan = opposite/adjacent, we get tan(θ) = 3/4. Choice B shows 4/3, which would be the reciprocal (cotangent, not tangent).
13. In a right triangle, the legs are $6$ (opposite $\theta$) and $8$ (adjacent $\theta$). What is $\sin(\theta)$?​
    Answer: $\frac{3}{5}$ For angle θ, the opposite side is 6, the adjacent side is 8, and we need to find the hypotenuse using the Pythagorean theorem: h² = 6² + 8² = 36 + 64 = 100, so h = 10. Using SOH: sin(θ) = opposite/hypotenuse = 6/10 = 3/5. Choice B (4/5) incorrectly uses the adjacent side instead of the opposite.
14. Which of the following equals $\cos\left(\frac{\pi}{3}\right)$?​
    Answer: $\frac{1}{2}$ The angle π/3 radians equals 60°. From the unit circle, cos(60°) = 1/2. This is a standard value to memorize. Choice A gives √3/2, which is actually sin(60°), not cos(60°).
15. What is $\sin(0^\circ)$?
    Answer: $0$ $\sin(0^\circ)$ is a standard unit circle value. Using SOH-CAH-TOA, sine represents the y-coordinate (or opposite/hypotenuse). For the angle $0^\circ$, $\sin(0^\circ)$ = 0. Choice B (1) is actually $\cos(0^\circ)$, not $\sin(0^\circ)$.
16. In the right triangle, if the opposite side is 6 and the adjacent side is 8, what is $\tan(\theta)$? Give your answer in simplest form.
    Answer: $\frac{3}{4}$ For angle θ, the opposite side is 6 and the adjacent side is 8. Using SOH-CAH-TOA, $tan(\theta) = opposite/adjacent$. Therefore, $tan(\theta) = 6/8 = 3/4$. Choice B ($\frac{4}{3}$) would be adjacent/opposite, which is the reciprocal relationship (cotangent).
17. What angle has $\cos(\theta) = \frac{1}{2}$?
    Answer: $60^\circ$ This asks for the angle where $\cos(\theta) = \frac{1}{2}$. From the unit circle, $\cos(60^\circ) = \frac{1}{2}$. Using SOH-CAH-TOA, cosine equals adjacent over hypotenuse, and $60^\circ$ is the standard angle with this ratio. Choice A ($30^\circ$) has $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, not $\frac{1}{2}$.
18. In a right triangle, an acute angle $\theta$ has opposite side length $9$ and adjacent side length $12$. What is $\tan(\theta)$?​
    Answer: $\frac{3}{4}$ For angle θ, the opposite side is 9 and the adjacent side is 12. Using TOA: tan(θ) = opposite/adjacent = 9/12. Simplifying this fraction by dividing both numerator and denominator by 3 gives us 3/4. Choice B (4/3) incorrectly uses adjacent/opposite instead of opposite/adjacent.
19. In a right triangle, the side opposite angle $\theta$ is $9$ and the side adjacent to $\theta$ is $12$. What is $\tan(\theta)$?
    Answer: $\frac{3}{4}$ For angle θ, the opposite side is 9 and the adjacent side is 12. Using SOH-CAH-TOA, tan(θ) = opposite/adjacent = 9/12 = 3/4. We can simplify this fraction by dividing both numerator and denominator by their greatest common factor of 3. Choice B gives 4/3, which would result from confusing the opposite and adjacent sides.
20. In the right triangle, if the adjacent side is 8 and the hypotenuse is 10, what is $\cos(\theta)$?
    Answer: $0.8$ For angle θ, the adjacent side is 8 and the hypotenuse is 10. Using SOH-CAH-TOA, cos(θ) = adjacent/hypotenuse. Therefore, cos(θ) = 8/10 = 0.8. Choice B (0.6) would be 6/10, which might confuse opposite over hypotenuse (sine), not cosine.

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