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AMC 12 only

Trigonometry Basics

The unit circle, special angles, and right-triangle ratios — the foundation every other trig lesson in this section builds on.

The unit circle definition

For an angle θ\theta measured from the positive x-axis, the point where the terminal ray meets the unit circle is exactly (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). This is the definition that extends sine and cosine beyond acute angles, to any real angle at all.

tan⁡θ=sin⁡θcos⁡θ,sin⁡2θ+cos⁡2θ=1\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \qquad \sin^2\theta+\cos^2\theta=1

Right-triangle ratios (SOH-CAH-TOA)

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}

For acute angles, this matches the unit-circle definition exactly — it’s just a restatement for a right triangle placed with one vertex at the origin.

Special angles and quadrant signs

Memorize these five values, and the rest of the unit circle follows from symmetry:

sin⁡(0°)=0, sin⁡(30°)=12, sin⁡(45°)=22, sin⁡(60°)=32, sin⁡(90°)=1\sin(0°)=0,\ \sin(30°)=\tfrac12,\ \sin(45°)=\tfrac{\sqrt2}{2},\ \sin(60°)=\tfrac{\sqrt3}{2},\ \sin(90°)=1

(Cosine values mirror these in reverse order, since cos⁡θ=sin⁡(90°−θ)\cos\theta=\sin(90°-\theta).) Beyond the first quadrant, find the reference angle (the acute angle to the nearest x-axis) and use the quadrant to determine sign: sine is positive in quadrants I and II, cosine is positive in I and IV, tangent is positive in I and III.

Worked example

Problem: Find the exact value of sin⁡(75°)\sin(75°).

Solution: Write 75°=45°+30°75° = 45°+30° and use the angle-addition formula (covered in the next lesson):

sin⁡(45°+30°)=sin⁡45°cos⁡30°+cos⁡45°sin⁡30°\sin(45°+30°) = \sin45°\cos30°+\cos45°\sin30°
=22⋅32+22⋅12=6+24= \dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2} + \dfrac{\sqrt2}{2}\cdot\dfrac12 = \dfrac{\sqrt6+\sqrt2}{4}

Practice problems

1.Find cos⁡(120°)\cos(120°) using the reference angle.

2.Find tan⁡(225°)\tan(225°).

3.A right triangle has an angle θ\theta with opposite side 5 and hypotenuse 13. Find cos⁡θ\cos\theta.

4.Find sin⁡(−30°)\sin(-30°).

Trig Identities & Equations