Reference Sheet · SAT Math · Geometry & Trig
Trigonometry
SOHCAHTOA, the complementary-angle identity, the Pythagorean identity, and the radian–degree relationship. SAT trig lives almost entirely inside right triangles. For full teaching, see the Geometry, Trigonometry & Systems notes.
SOHCAHTOA — the three ratios
Pick one acute angle of a right triangle to be your reference angle, $\theta$. Then the three sides have these names:
Sine
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
S-O-H: Sine = Opposite over Hypotenuse.
EXAMPLE
Right triangle: opposite $= 3$, hypotenuse $= 5$. Find $\sin\theta$.
$\sin\theta = \dfrac{3}{5} = $ $0.6$
Cosine
$$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
C-A-H: Cosine = Adjacent over Hypotenuse.
EXAMPLE
Right triangle: adjacent $= 4$, hypotenuse $= 5$. Find $\cos\theta$.
$\cos\theta = \dfrac{4}{5} = $ $0.8$
Tangent
$$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$
T-O-A: Tangent = Opposite over Adjacent. Equivalently, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
EXAMPLE
Right triangle: opposite $= 3$, adjacent $= 4$. Find $\tan\theta$.
$\tan\theta = \dfrac{3}{4} = $ $0.75$
Try it yourself
- Find the slider for $\theta$ (theta) in the panel on the left.
- Drag it to change the angle (in degrees).
- Watch the red point trace around the circle as $\theta$ changes.
Notice: The x-coordinate of the point is $\cos\theta$; the y-coordinate is $\sin\theta$. That's why the unit circle is the bridge between angles and SOHCAHTOA — it extends those ratios to all angles, not just acute ones.
Trig identities
Complementary-angle identity
$$\sin\theta = \cos(90° - \theta)$$
The sine of an angle equals the cosine of its complement. The SAT loves this — questions ask "if $\sin x = 0.3$, what is $\cos(90° - x)$?" The answer is the same value.
EXAMPLE
If $\sin(40°) = 0.643$, find $\cos(50°)$.
$50°$ is the complement of $40°$, so $\cos(50°) = \sin(40°) = $ $0.643$
Pythagorean identity
$$\sin^2\theta + \cos^2\theta = 1$$
Comes directly from $a^2 + b^2 = c^2$ applied to the unit circle. Use when given one ratio and asked for another.
EXAMPLE
If $\sin\theta = 0.6$ and $\theta$ is acute, find $\cos\theta$.
$0.36 + \cos^2\theta = 1$
$\cos^2\theta = 0.64 \ \Rightarrow \ \cos\theta = $ $0.8$
Radians
Degree-radian conversion
$$180° = \pi \text{ radians}$$
A radian is just another unit for measuring angles, like Celsius vs Fahrenheit. To convert: multiply by $\dfrac{\pi}{180°}$ for degrees → radians, or by $\dfrac{180°}{\pi}$ for radians → degrees.
EXAMPLE
Convert $60°$ to radians.
$60° \cdot \dfrac{\pi}{180°} = \dfrac{60\pi}{180} = $ $\dfrac{\pi}{3}$
Common values to know
$$30° = \tfrac{\pi}{6},\ \ 45° = \tfrac{\pi}{4},\ \ 60° = \tfrac{\pi}{3}$$$$90° = \tfrac{\pi}{2},\ \ 180° = \pi,\ \ 360° = 2\pi$$
Memorize these. They show up constantly on the SAT and AP-level problems.
EXAMPLE
Convert $\dfrac{3\pi}{4}$ radians to degrees.
$\dfrac{3\pi}{4} \cdot \dfrac{180°}{\pi} = \dfrac{3 \cdot 180°}{4} = $ $135°$