Trigonometry

The three trig ratios in a right triangle (with respect to one of the non-right angles, $\theta$):

• Sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

SOH-CAH-TOA mnemonic:

• SOH: Sine = Opposite over Hypotenuse.
• CAH: Cosine = Adjacent over Hypotenuse.
• TOA: Tangent = Opposite over Adjacent.

Reading the triangle:

• The hypotenuse is opposite the right angle (always the longest side).
• The opposite is opposite to the angle $\theta$.
• The adjacent is next to the angle $\theta$ (not the hypotenuse).

Swap angles and the opposite and adjacent swap with them.

The two key SAT-tested values:

These come straight from the special right triangles (30-60-90 and 45-45-90).

Complementary angle identity (commonly tested):

$\sin(90° - \theta) = \cos\theta$

$\cos(90° - \theta) = \sin\theta$

If $\sin(35°) = 0.574$, then $\cos(55°) = 0.574$ — same value, complementary angles.

This is why we say cosine — co + sine, where 'co' refers to complementary angle.

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$. (Less commonly tested but worth knowing.)

Inverse trig. If you know a ratio and want the angle, use inverse:

• $\sin^{-1}(0.5) = 30°$ (since $\sin 30° = 0.5$).
• $\cos^{-1}(\sqrt{3}/2) = 30°$.
• $\tan^{-1}(1) = 45°$.

The SAT calculator can compute these directly.

SAT-typical setups:

• "In right triangle ABC with $\angle B = 90°$, $\sin A = 0.6$. Find $\cos A$." → Use Pythagorean identity, or recognize the 3-4-5 triangle.
• "What is $\cos(40°)$ if $\sin(50°) = 0.766$?" → complementary identity: $\cos(40°) = \sin(50°) = 0.766$.

Reference triangle. When solving for an angle's sine or cosine, draw the right triangle with the given angle. Label the sides using the ratio definitions, then read off whichever side ratio you need.