🌀Trig Identities

Last lesson, sin, cos and tan looked like three unrelated table entries. They are actually one family: for any given angle, the three ratios satisfy equations that hold always and everywhere — trig identities. The most famous one is just the Pythagorean theorem in different clothes. In this lesson we derive it straight from geometry, then use identities to simplify an expression.

sin and cos on the unit circle

Draw a circle of radius 1 (the unit circle) centered at the origin. From the center, draw a radius making an angle $\theta$ with the x-axis, meeting the circle at a point. That point has coordinates $(\cos\theta, \sin\theta)$ — cosine is the horizontal coordinate, sine the vertical one.

Why? Complete the triangle: the radius is the hypotenuse, and the horizontal and vertical segments are the legs. The radius is 1, so it is the hypotenuse; the leg adjacent to $\theta$ is the horizontal coordinate and the leg opposite $\theta$ is the vertical one. The definitions of trigonometry divide by the hypotenuse — and dividing by 1 changes nothing, so the ratios simply are the coordinates. The unit circle upgrades trigonometry from a property of triangles to the position of a point, and it settles an old question along the way: why do sin and cos never leave the range from −1 to 1? A coordinate's absolute value never exceeds the point's distance from the center — and on a radius-1 circle, that distance is exactly 1.

The Pythagorean theorem's other name

The point lies on the unit circle, so its distance from the origin is always 1. Apply the distance formula (which is the Pythagorean theorem) to $(x, y)$: $x^2 + y^2 = 1$. Substitute $x = \cos\theta$ and $y = \sin\theta$:

$\sin^2\theta + \cos^2\theta = 1$

This is the Pythagorean identity, and it holds for every angle. Check it against last lesson's table: at $\theta = 30^\circ$, $0.5^2 + 0.866^2 \approx 0.25 + 0.75 = 1$; at $\theta = 45^\circ$, $0.707^2 + 0.707^2 \approx 0.5 + 0.5 = 1$. No exceptions — every angle's endpoint lands on the unit circle.

Leg a3Leg b4

3² + 4² = 9 + 16 = 25 = c²

Hypotenuse c c = √25 ≈ 5.00整数勾股数 · Pythagorean triple!

Drag the two legs and watch the areas of the three squares. Keep one eye on this: the square on the hypotenuse equals the two squares on the legs combined. Shrink the hypotenuse to 1 — its square has area 1, and the two leg squares are exactly $\cos^2\theta$ and $\sin^2\theta$. The identity is the Pythagorean theorem living on the unit circle.

The quotient identity: where tan comes from

Tan has a definition of its own — can it be written using sin and cos? Divide the numerator and denominator of "opposite ÷ adjacent" by the hypotenuse; the ratio does not change, and what remains is two familiar ratios dividing each other:

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

This quotient identity matches the table too: $\tan 45^\circ = 0.707 \div 0.707 = 1$, and $\tan 30^\circ = 0.5 \div 0.866 \approx 0.577$. Its meaning goes further — of the three ratios, only two are independent. The third is decided by the first two: know sin and cos, and tan comes free.

Simplifying with identities

An identity is a "rearrangement permit": its two sides are always equal, so you may swap them at will. Simplifying means rewriting an expression into an easier form.

In Sine and Cosine Rules, these ratios step outside right triangles to tackle oblique ones.

Check yourself

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Quick quiz

↺ Try again

1. 1. At θ = 45°, sin²θ + cos²θ equals?

   0.51about 1.41

2. 2. An acute angle θ satisfies sinθ = 0.6. What is cosθ?

   0.40.80.64

3. 3. Simplify 1 − sin²θ.

   cos²θsinθ1 − cos²θ