This is a trigonometry question requiring derivation of multiple trig ratios from one. Choice C (31/20) is correct — sin(θ) = 3/5 means the triangle has opposite = 3 and hypotenuse = 5, so it's a 3-4-5 right triangle with adjacent = 4. Therefore: cos(θ) = 4/5 and tan(θ) = 3/4. Sum: 4/5 + 3/4 = 16/20 + 15/20 = 31/20. Choice A (1) likely invokes the Pythagorean identity sin²θ + cos²θ = 1, confusing that identity with the sum cos(θ) + tan(θ). Choice B (7/5) comes from computing 4/5 + 3/5 = 7/5 — correctly finding cos but computing tan as 3/5 (using the hypotenuse instead of the adjacent side in the denominator). Choice D (2) may come from estimating both ratios as approximately 1 and adding. Pro tip: If sin(θ) = a/c, immediately sketch a right triangle with opposite = a, hypotenuse = c, and use the Pythagorean theorem to find the adjacent side. From there, cos and tan follow directly.

The correct answer is D (4). The period of f(x) = A sin(bx) + c is given by 2π/b. Here b = π/2. Period = 2π ÷ (π/2) = 2π × (2/π) = 4. A (π/2) reports the b-value itself as the period rather than computing 2π/b. B (2) results from computing 2π/b with b = π (misreading the coefficient as π instead of π/2): 2π/π = 2. C (3) reports the amplitude coefficient rather than the period — confusing the A and b parameters. Pro tip: the period formula is 2π/b where b is the coefficient multiplying x, not x itself.

This is a trigonometry question testing SOHCAHTOA applied to a labeled right triangle. Choice B (8/15) is correct — from angle C's perspective: the opposite side is AB = 8 and the adjacent side is BC = 15. tan C = opposite/adjacent = 8/15. (The hypotenuse = √(8² + 15²) = √289 = 17.) Choice A (8/17) gives sin C, not tan C — it correctly identifies opposite = 8 but uses the hypotenuse (17) as the denominator instead of the adjacent side. Choice C (15/17) gives cos C — adjacent over hypotenuse. Choice D (15/8) gives tan B, the other acute angle — it swaps opposite and adjacent, giving the tangent from the perspective of angle B rather than angle C. Pro tip: Before writing a trig ratio, identify which angle you're evaluating and label each side (opposite, adjacent, hypotenuse) relative to THAT angle. The same side can be "opposite" for one angle and "adjacent" for another.

The angle 60° is a special angle from the unit circle and 30-60-90 triangles. Using these fundamental trigonometric values, sin(60°) = √3/2. This is a standard result that should be memorized along with other special angle values. Choice A gives 1/2, which is actually sin(30°), not sin(60°).

For angle θ, the adjacent side = 12 and the hypotenuse = 13. Using CAH: cos = adjacent/hypotenuse, we get cos(θ) = 12/13. Choice B shows 5/13, which would be sin(θ) using the opposite side length of 5.

For angle θ, the opposite side = 8 and the hypotenuse = 17. Using SOH: sin = opposite/hypotenuse, we get sin(θ) = 8/17. Choice A shows 15/17, which would be cos(θ) using the adjacent side length of 15.

In the unit circle, $\pi/4$ radians equals $45^\circ$. Using SOH-CAH-TOA, tangent represents opposite over adjacent. For the special angle $\pi/4$ ($45^\circ$), $\tan(\pi/4) = 1$. Choice C ($\sqrt{3}$) is actually $\tan(\pi/3)$ or $\tan(60^\circ)$.

In the unit circle, $\pi/6$ radians equals $30^\circ$. Using SOH-CAH-TOA, sine represents the y-coordinate (or opposite/hypotenuse). For the special angle $\pi/6$ ($30^\circ$), $sin(\pi/6) = 1/2$. Choice B ($\sqrt{3}/2$) is actually $sin(\pi/3)$ or $sin(60^\circ$).

90° is where the unit circle intersects the positive y-axis. Using SOH-CAH-TOA, sin(90°) = opposite/hypotenuse = 1. At 90°, the y-coordinate reaches its maximum value. Choice B gives 0, which is sin(0°), not sin(90°).

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°).