Questions: Sine, Cosine, and Tangent Ratios

Question 1

In a right triangle with angle A = 40° and hypotenuse = 15 cm, which calculation correctly finds the side adjacent to angle A?

• A. 15 × sin(40°)
• B. 15 × cos(40°)
• C. 15 × tan(40°)
• D. 15 / tan(40°)

Answer: (B)

cos(A) = adjacent/hypotenuse, so adjacent = hypotenuse × cos(A) = 15 × cos(40°). Option A uses sine, which gives the opposite side (sin = opposite/hyp). Option C uses tangent (opposite/adjacent), which requires knowing a leg rather than the hypotenuse. Matching the ratio to the sides you know and want is the core skill.

Question 2

In a right triangle, the side labeled 'opposite' when working from acute angle A becomes which of the following when working from the other acute angle B?

• A. The hypotenuse
• B. Still the opposite side — the label doesn't change
• C. The adjacent side
• D. It becomes undefined — you can only designate one reference angle at a time

Answer: (C)

The labels 'opposite' and 'adjacent' are not fixed properties of the sides — they shift depending on which angle is the reference. The side facing angle A is its 'opposite'; that same side runs along angle B toward the right angle, making it 'adjacent' for B. This is the most disorienting aspect of trig ratios and the source of most labeling errors. Always re-label after switching reference angles.

Question 3

The hypotenuse appears as the denominator in both sine and cosine — regardless of which acute angle you choose as the reference angle.

• T. True
• F. False

Answer: True

The hypotenuse is always opposite the right angle and is always the longest side. sin(A) = opposite/hypotenuse and cos(A) = adjacent/hypotenuse — both use the hypotenuse as the denominator. Tangent (opposite/adjacent) is the only ratio that never involves the hypotenuse. This is consistent no matter which acute angle you designate as A.

Question 4

sin⁻¹(0.6) means the same thing as 1/sin(0.6).

• T. True
• F. False

Answer: False

sin⁻¹ is the inverse sine function — it takes a ratio and returns the angle whose sine equals that ratio. It does NOT mean the reciprocal. The reciprocal of sine is cosecant (csc). The superscript −1 denotes function inversion, not arithmetic reciprocal. Confusing these is a common mistake: sin⁻¹(0.6) ≈ 36.9°, while 1/sin(0.6) ≈ 1.77 (radians context) — completely different values.

Question 5

In a right triangle where you know two sides but not the angle, describe the process for finding the missing angle using a trig ratio.

Model answer: Identify the two known sides relative to the desired angle, labeling them (opposite, adjacent, or hypotenuse). Compute the appropriate ratio: sin if you have opposite and hypotenuse, cos if adjacent and hypotenuse, tan if opposite and adjacent. Then apply the corresponding inverse function: angle = sin⁻¹(ratio), cos⁻¹(ratio), or tan⁻¹(ratio).

The inverse function 'undoes' the trig ratio, converting a ratio back into an angle. For example, if opposite = 5 and hypotenuse = 10, then sin(A) = 0.5, so A = sin⁻¹(0.5) = 30°. The key is choosing which ratio links your two known sides — that determines which inverse function to apply.