Basic Trigonometry

1. Introduction

SAT trigonometry is far narrower than a full trig course, which is excellent news: you only need right-triangle ratios, one complementary-angle identity, the special-angle values, and a little radian awareness. There are no proofs, no law of sines, and rarely any graphing of trig functions. If you internalize SOH-CAH-TOA and the fact that sine and cosine of complementary angles are equal, you can handle the overwhelming majority of SAT trig questions.

The recurring difficulty is labeling. "Opposite" and "adjacent" are defined relative to the angle you choose, so the same side can be opposite for one angle and adjacent for another. A second classic stumbling block is calculator mode — degrees vs. radians — which silently produces wrong answers if set incorrectly.

The SAT also connects trigonometry to special right triangles, the unit circle (at a basic level), and word problems involving angles of elevation and depression. You may be asked to find a side length, compute a ratio, or solve for an angle — always in the context of a right triangle. This article builds the right-triangle ratios from the ground up, connects them to the special right triangles you already know from geometry, drills the complementary-angle identity the SAT loves, and lays out the calculator and timing tactics that turn these into quick points.

What the SAT actually tests. Trigonometry is a small slice of Additional Topics — typically one or two questions per test. The most common types are: find a side using SOH-CAH-TOA, apply the complementary identity, compute a ratio from side lengths, and use special-angle exact values. Inverse trig (finding an angle) appears on the calculator section. There is no graphing of sine or cosine curves.

Study priority. Memorize SOH-CAH-TOA and the complementary identity first — together they cover most questions. Special-angle values ($30^\circ$, $45^\circ$, $60^\circ$) are the second priority. The Pythagorean identity is a useful backup for finding one ratio from another.

Diagnostic checklist. Before submitting: Did I label opposite, adjacent, and hypotenuse from the correct angle? Is my calculator in degree mode (unless radians are specified)? Did I use the complementary identity when angles sum to $90^\circ$? For the Pythagorean identity, did I take the positive root for an acute angle?

2. Core Concepts

2.1 The Right Triangle and Its Sides

In a right triangle, the hypotenuse is the longest side, always opposite the $90^\circ$ angle. Choose one of the two acute angles, call it $\theta$. Relative to $\theta$:

• the opposite side is the leg across from $\theta$,
• the adjacent side is the leg next to $\theta$ (that isn't the hypotenuse).

Switching your attention to the other acute angle swaps which leg is "opposite" and which is "adjacent" — the hypotenuse never changes.

2.2 SOH-CAH-TOA

The three primary ratios for an acute angle $\theta$:

• Sine: $\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$ (SOH)
• Cosine: $\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$ (CAH)
• Tangent: $\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$ (TOA)

Note that $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, and tangent never involves the hypotenuse.

2.3 The Complementary Angle Relationship

In any right triangle the two acute angles sum to $90^\circ$ — they are complementary. Because the opposite side of one acute angle is the adjacent side of the other, sine and cosine trade places:

$\sin\theta = \cos(90^\circ - \theta) \quad\text{and}\quad \cos\theta = \sin(90^\circ - \theta).$

This is the single most-tested trig identity on the SAT. If $\sin x^\circ = 0.6$, then $\cos(90^\circ - x)^\circ = 0.6$ instantly — no triangle needed.

2.4 Special-Angle Exact Values

From the $45$-$45$-$90$ and $30$-$60$-$90$ triangles:

• $\sin 30^\circ = \dfrac{1}{2}$, $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$, $\tan 30^\circ = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$.
• $\sin 45^\circ = \cos 45^\circ = \dfrac{\sqrt{2}}{2}$, $\tan 45^\circ = 1$.
• $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$, $\cos 60^\circ = \dfrac{1}{2}$, $\tan 60^\circ = \sqrt{3}$.

Notice $\sin 30^\circ = \cos 60^\circ$ — the complementary relationship in action.

2.5 Radian Measure

Angles can be measured in radians instead of degrees, where a full circle is $2\pi$ radians and a straight angle is $\pi$ radians ($180^\circ = \pi$). Convert by multiplying:

• degrees $\to$ radians: multiply by $\dfrac{\pi}{180}$,
• radians $\to$ degrees: multiply by $\dfrac{180}{\pi}$.

So $60^\circ = 60 \cdot \frac{\pi}{180} = \frac{\pi}{3}$ radians.

2.6 The Pythagorean Identity

For any angle $\theta$:

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

This follows from the Pythagorean theorem applied to a unit circle or any right triangle with hypotenuse $1$. On the SAT, it lets you find $\cos\theta$ given $\sin\theta$ (or vice versa) for an acute angle without drawing a triangle.

2.7 Angles of Elevation and Depression

An angle of elevation is measured upward from the horizontal to a line of sight. An angle of depression is measured downward from the horizontal. Both create right triangles with the horizontal, and the SAT uses them in height-and-distance word problems. The angle of depression from one point equals the angle of elevation from the other (alternate interior angles).

2.8 Cofunction Identities in SAT Language

The complementary-angle identity is sometimes written using cofunction language: sine and cosine are cofunctions, as are tangent and cotangent. The SAT does not use this vocabulary, but the idea is the same: $\sin(90^\circ - \theta) = \cos\theta$. When a problem gives $\cos(35^\circ)$ and asks for $\sin(55^\circ)$, the answer is immediate.

2.9 The Unit Circle Connection (Basic)

On a unit circle (radius $1$), the coordinates of a point at angle $\theta$ are $(\cos\theta, \sin\theta)$. The SAT does not require unit-circle fluency, but it explains why the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ holds: the point satisfies $x^2 + y^2 = 1$, and $x = \cos\theta$, $y = \sin\theta$.

2.10 When to Use Each Ratio

Choosing the correct ratio saves time and prevents errors:

• Know opposite and hypotenuse $\to$ use sine.
• Know adjacent and hypotenuse $\to$ use cosine.
• Know opposite and adjacent $\to$ use tangent.
• Know sine and need cosine (acute angle) $\to$ use $\sin^2\theta + \cos^2\theta = 1$.
• See $\sin$ and $\cos$ of angles summing to $90^\circ$ $\to$ use the complementary identity.

2.11 Word Problem Setup for Trig

For angle-of-elevation or depression problems:

1. Draw a right triangle with the horizontal as one leg.
2. Mark the angle of elevation or depression at the correct vertex.
3. Label the known side (often horizontal distance or height) and the unknown.
4. Choose SOH, CAH, or TOA based on which sides are known and which is needed.

The horizontal line and the line of sight are always perpendicular, guaranteeing a right triangle.