Trigonometry Basics (SOH CAH TOA)

SOH CAH TOA — The Foundation

The cornerstone of right-triangle trigonometry is the mnemonic SOH CAH TOA:

• SOH: $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• CAH: $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• TOA: $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

These ratios apply to right triangles only. The "opposite" and "adjacent" sides are defined relative to the specific angle $\theta$ you are working with — they change depending on which angle you choose. The hypotenuse is always the longest side, directly across from the right angle.

How to identify the sides:

1. Find the angle $\theta$ in question (not the right angle)
2. The side directly across from $\theta$ is the opposite side
3. The side next to $\theta$ (that is not the hypotenuse) is the adjacent side
4. The side across from the $90°$ angle is always the hypotenuse

The Reciprocal Ratios

The ACT occasionally tests the three reciprocal trig functions:

• $\csc\theta = \frac{1}{\sin\theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$ (cosecant)
• $\sec\theta = \frac{1}{\cos\theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$ (secant)
• $\cot\theta = \frac{1}{\tan\theta} = \frac{\text{Adjacent}}{\text{Opposite}}$ (cotangent)

You do not need to memorize separate formulas for these — just remember that each is the reciprocal (the "flip") of a primary ratio. Cosecant goes with sine, secant goes with cosine, and cotangent goes with tangent.

Finding Missing Sides

When you know one acute angle (besides the right angle) and one side of a right triangle, you can find any other side using SOH CAH TOA.

Setup process:

1. Label the sides relative to the known angle: Opposite (O), Adjacent (A), Hypotenuse (H)
2. Identify which two sides are involved — the one you know and the one you want to find
3. Choose the trig ratio that connects those two sides
4. Set up the equation and solve

Detailed example: In a right triangle where $\theta = 35°$ and the hypotenuse is 12:

• To find the opposite side: you need O and H → use SOH: $\sin 35° = \frac{\text{opp}}{12}$, so opp $= 12\sin 35° \approx 12(0.5736) \approx 6.88$
• To find the adjacent side: you need A and H → use CAH: $\cos 35° = \frac{\text{adj}}{12}$, so adj $= 12\cos 35° \approx 12(0.8192) \approx 9.83$

Finding Missing Angles

When you know two sides of a right triangle, you can find an angle using inverse trig functions (also called arc-functions):

$\theta = \sin^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right) \quad \text{or} \quad \theta = \cos^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right) \quad \text{or} \quad \theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)$

Important notation: $\sin^{-1}(x)$ means "the angle whose sine is $x$" (also written $\arcsin x$). It does not mean $\frac{1}{\sin x}$. The reciprocal of sine is cosecant, not inverse sine.

The Unit Circle Basics

The unit circle is a circle of radius 1 centered at the origin. An angle $\theta$ measured counterclockwise from the positive $x$-axis corresponds to the point $(\cos\theta, \sin\theta)$ on the circle. This means the $x$-coordinate is cosine and the $y$-coordinate is sine.

Key angles to memorize (you need these for the ACT):

Degrees	Radians	$\sin$	$\cos$	$\tan$
$0°$	$0$	$0$	$1$	$0$
$30°$	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45°$	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90°$	$\frac{\pi}{2}$	$1$	$0$	undefined

Memory trick: For sine values at 0°, 30°, 45°, 60°, 90°, the pattern is $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$. For cosine, the same values appear in reverse order.

Signs by Quadrant (ASTC)

Use the mnemonic "All Students Take Calculus" to remember which trig functions are positive in each quadrant:

• Quadrant I: All positive (sin+, cos+, tan+)
• Quadrant II: Sine positive only (sin+, cos−, tan−)
• Quadrant III: Tangent positive only (sin−, cos−, tan+)
• Quadrant IV: Cosine positive only (sin−, cos+, tan−)

The ACT tests this directly: "If $\sin\theta > 0$ and $\cos\theta < 0$, in which quadrant does $\theta$ lie?" The answer is Quadrant II (where sine is positive and cosine is negative).

Special Right Triangles

Two special right triangles appear on virtually every ACT:

45-45-90 triangle: sides in ratio $1 : 1 : \sqrt{2}$. Both legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg. This triangle comes from cutting a square along its diagonal.

30-60-90 triangle: sides in ratio $1 : \sqrt{3} : 2$. The shortest side is opposite the $30°$ angle, the medium side ($\sqrt{3}$) is opposite the $60°$ angle, and the longest side (2) is the hypotenuse. This triangle comes from cutting an equilateral triangle in half.

Scaling: These ratios can be scaled by any factor. A 30-60-90 triangle with shortest side 5 has sides $5, 5\sqrt{3}, 10$.

Fundamental Identities

The ACT may test these basic identities:

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$

This is the most important trig identity. It can be rearranged: $\sin^2\theta = 1 - \cos^2\theta$ or $\cos^2\theta = 1 - \sin^2\theta$.

Tangent identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

These two identities, combined with SOH CAH TOA, cover the vast majority of ACT trig questions.

Radians and Degrees

Some ACT questions use radians instead of degrees. The key conversion:

$\pi \text{ radians} = 180°$

So $1 \text{ radian} = \frac{180°}{\pi} \approx 57.3°$, and to convert degrees to radians, multiply by $\frac{\pi}{180}$.

Common conversions: $30° = \frac{\pi}{6}$, $45° = \frac{\pi}{4}$, $60° = \frac{\pi}{3}$, $90° = \frac{\pi}{2}$, $180° = \pi$, $360° = 2\pi$.

Tip 1: Draw and Label the Triangle

For every right-triangle trig problem, draw the triangle and clearly label all known sides and angles. Mark which side is opposite, which is adjacent, and which is the hypotenuse relative to the angle in question. This single step prevents the most common errors.

Tip 2: Check Your Calculator Mode

Make sure your calculator is in degree mode (not radians) unless the problem explicitly uses radians. This is a devastating and common source of errors. On TI calculators, press MODE and verify the angle setting. If you enter $\sin(30)$ expecting 0.5 but your calculator is in radians, you will get $-0.988$ — a completely wrong answer.

Tip 3: Memorize the Special Triangles

Many ACT trig questions specifically use 30-60-90 or 45-45-90 triangles. Knowing the side ratios by heart lets you solve these problems without a calculator, saving 30–60 seconds per question.

Tip 4: Eliminate Answer Choices Using Bounds

If you know that $\sin\theta$ must be between 0 and 1 for an acute angle, you can immediately eliminate any answer choice outside that range. Similarly, $\cos\theta$ is between $-1$ and $1$ for all angles. Use these bounds to narrow your choices.

Tip 5: Watch for Radians vs. Degrees

If a problem gives an angle as $\frac{\pi}{6}$, that is $30°$. If you see a naked number like 2 as an angle, it probably means 2 radians (about $114.6°$), not $2°$. Context clues and the presence of $\pi$ help you determine which unit is being used.