One ratio gives all six
Use the identities and the reciprocals to find every ratio from one.
The missing ratios
A student knows only that tan A = 1√3, for an acute angle A.
She does not have a triangle, a protractor or a calculator. Yet the chapter says every other ratio of A is fixed.
Can she find sin A, cos A and the other ratios from tan A alone?
What this lesson covers
The idea
Using the trigonometric identities, each trigonometric ratio can be expressed in terms of the other ratios, so if any one ratio is known the values of the others can be determined.
The missing ratios
A student knows only that tan A = 1/√3, for an acute angle A.
She does not have a triangle, a protractor or a calculator. Yet the chapter says every other ratio of A is fixed.
Can she find sin A, cos A and the other ratios from tan A alone?
- Yes, using the identities and the reciprocals
- No, she also needs a side length
- Only the reciprocal cot A
Climb down the ladder
Start from the given ratio. At each step, pick the rule that gives the next ratio. The card fills in with its exact value.
Each ratio through the others
Using the trigonometric identities, each ratio can be expressed in terms of the other ratios. If any one ratio is known, the values of the others can be determined.
Example 9 of your book writes everything in terms of sin A: cos² A = 1 − sin² A, so cos A = √(1 − sin² A) (the positive root, because A is acute). Then tan A = sin A / √(1 − sin² A) and sec A = 1 / √(1 − sin² A).
Notes
From any one ratio you can find all the others: use the reciprocals, sin A / cos A = tan A, and the identities cos² A + sin² A = 1, 1 + tan² A = sec² A, cot² A + 1 = cosec² A.
Check yourself
A is acute and cos A = 3/5. Find sin A as a decimal.
Answer: 0.8
sin² A = 1 − cos² A = 1 − 9/25 = 16/25, so sin A = 4/5 = 0.8.
A is acute and sin A = 8/17. Find cos A as a decimal, to three places.
Answer: 0.882
cos² A = 1 − 64/289 = 225/289, so cos A = 15/17, about 0.882.
A is acute. Which expresses tan A in terms of sin A?
A is acute and tan A = 1. Find sin A as a decimal, to two places.
Answer: 0.71
sec² A = 1 + 1 = 2, so sec A = √2 and cos A = 1/√2. Then sin² A = 1 − 1/2 = 1/2 and sin A = 1/√2, about 0.71. (This is A = 45°.)
- sin A / (1 − sin² A). The bottom must be cos A, which is √(1 − sin² A), the square root.
- √(1 − sin² A) / sin A. This is cos A / sin A, which is cot A.
- sin A / √(1 − sin² A) — correct. Yes! tan A = sin A / cos A and cos A = √(1 − sin² A).