Trigonometric ratios and fundamental identities

Maths · Fundamentals of trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Put the angle θ in standard position and take any point P(x, y) on its terminal side, at distance r = √(x² + y²) from the origin. The six trigonometric functions are ratios of x, y and r. Because r is always positive, the sign of each function depends only on the signs of x and y, that is, on the quadrant.

You must know the exact values for 30°, 45° and 60°, and for the quadrantal angles 0°, 90°, 180° and 270°.

The three fundamental identities connect the functions. If you know one ratio and the quadrant, they give you all the others.

Rules to remember

sin θ = y/r, cos θ = x/r, tan θ = y/x; cosec θ = r/y, sec θ = r/x, cot θ = x/y
Signs: quadrant I all positive; II only sin and cosec; III only tan and cot; IV only cos and sec
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
(cos θ, sin θ) is (1, 0) at 0°, (0, 1) at 90°, (−1, 0) at 180° and (0, −1) at 270°
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Common mistake

After taking a square root, students keep the positive value without checking the quadrant. Decide the sign from the quadrant before you write the answer.

Worked example

sin θ = 8/17 and the terminal side of θ is in the second quadrant. Find cos θ and tan θ.

cos²θ = 1 − sin²θ = 1 − 64/289 = 225/289, so cos θ = ±15/17.
In the second quadrant cos θ is negative: cos θ = −15/17.
tan θ = sin θ ÷ cos θ = (8/17) ÷ (−15/17) = −8/15.

Answer: cos θ = −15/17 and tan θ = −8/15.

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