Basic Trigonometry (sin, cos, tan) — SAT Math Explained
The three primary trigonometric ratios — sine, cosine, and tangent — express relationships between an acute angle in a right triangle and the ratios of its sides.
The Core Idea
For any given angle in a right triangle, the ratio of specific pairs of sides is always the same, regardless of the triangle's size. This universality makes trig ratios tools for finding any side or angle in a right triangle.
Definitions
sin(θ) = Opposite / Hypotenuse — the side ACROSS from the angle over the hypotenuse
cos(θ) = Adjacent / Hypotenuse — the side NEXT TO the angle over the hypotenuse
tan(θ) = Opposite / Adjacent — the side across over the side next to
Mnemonic S O H C A H T O A
SOH: Sine = Opposite/Hypotenuse; CAH: Cosine = Adjacent/Hypotenuse; TOA: Tangent = Opposite/Adjacent
Identifying Sides
The HYPOTENUSE is always opposite the right angle — the longest side
OPPOSITE is the side directly across from the angle you're using
ADJACENT is the side next to the angle (not the hypotenuse)
Using Trig To Find Sides
1. Identify the angle you're using
2. Identify which sides are involved (opposite, adjacent, or hypotenuse)
3. Choose the correct trig ratio (sin, cos, or tan)
4. Set up the equation: trig(angle) = side/side
5. Solve for the unknown side using multiplication or division
Using Inverse Trig To Find Angles
sin⁻¹, cos⁻¹, tan⁻¹ (also written arcsin, arccos, arctan) — used to find the angle when you know the side ratio
Special Angles
sin=1/2, cos=√3/2, tan=1/√3
sin=√2/2, cos=√2/2, tan=1
sin=√3/2, cos=1/2, tan=√3
Common Errors to Avoid
Identifying adjacent and opposite incorrectly (they depend on which angle you're using)
Using the trig ratio for the wrong angle
Forgetting that inverse trig (sin⁻¹) gives an angle, not a ratio
Practice: Basic Trigonometry (sin, cos, tan)
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In a right triangle, sin θ equals:
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