ACT Trigonometry: Ratios, Angles, Equations, Unit Circle, and Common Traps

ACT trigonometry is usually built from a small number of durable ideas:

$$\boxed{\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}}$$ $$\boxed{\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}}$$ $$\boxed{\tan\theta=\frac{\text{opposite}}{\text{adjacent}}}$$

This is SOH-CAH-TOA.

The current ACT Math description places trigonometric ratios inside Geometry, one of the largest Math categories at roughly 17–20%.

The key is not to memorize random triangles. It is to identify:

1. reference angle;
2. opposite/adjacent/hypotenuse;
3. correct trig ratio;
4. whether the problem is triangle-based, unit-circle-based, or equation-based.

Identify the hypotenuse first

In a right triangle, the hypotenuse is:

• opposite the $90^\circ$ angle;
• always the longest side.

It does not depend on which acute angle you are using.

Then "opposite" and "adjacent" depend on the reference angle.

Example: solve a side

Suppose:

• $\theta=35^\circ$;
• adjacent side = 10;
• need opposite side $x$.

Use tangent:

$$\tan35^\circ=\frac{x}{10}$$

So:

$$x=10\tan35^\circ$$

Calculator gives the numerical value.

Example: solve an angle

If:

• opposite = 6;
• adjacent = 8;

then:

$$\tan\theta=\frac68=\frac34$$

So:

$$\theta=\tan^{-1}\left(\frac34\right)$$

Use inverse trig.

Inverse trig does not mean reciprocal

$$\tan^{-1}(x)$$

means inverse tangent, not:

$$\frac1{\tan x}$$

The reciprocal of tangent is cotangent:

$$\cot x=\frac1{\tan x}$$

This notation trap matters.

Special right triangles

45-45-90

Side ratio:

$$\boxed{1:1:\sqrt2}$$

If leg = 5:

$$\text{hypotenuse}=5\sqrt2$$

30-60-90

Opposite $30^\circ$:

$$x$$

Opposite $60^\circ$:

$$x\sqrt3$$

Hypotenuse:

$$2x$$

Ratio:

$$\boxed{1:\sqrt3:2}$$

These are faster than calculator trig when exact values are expected.

Pythagorean theorem connection

For right triangle:

$$a^2+b^2=c^2$$

Sometimes you need to find a missing side before applying trig.

Example: legs 5 and 12.

Hypotenuse:

$$13$$

Then for angle opposite side 5:

$$\sin\theta=\frac5{13}$$

Unit circle coordinates

On unit circle:

$$\boxed{(\cos\theta,\sin\theta)}$$

This is the bridge between triangle trig and graph/function trig.

At:

$$0^\circ$$

point:

$$(1,0)$$

so:

$$\cos0^\circ=1,\quad\sin0^\circ=0$$

Key unit-circle values

30°

$$\cos30^\circ=\frac{\sqrt3}{2}$$ $$\sin30^\circ=\frac12$$

45°

$$\cos45^\circ=\frac{\sqrt2}{2}$$ $$\sin45^\circ=\frac{\sqrt2}{2}$$

60°

$$\cos60^\circ=\frac12$$ $$\sin60^\circ=\frac{\sqrt3}{2}$$

These come directly from special triangles.

Signs by quadrant

Mnemonic:

All Students Take Calculus

Quadrant I:

• all positive.

Quadrant II:

• sine positive.

Quadrant III:

• tangent positive.

Quadrant IV:

• cosine positive.

For example:

$$150^\circ$$

is Quadrant II with reference angle:

$$30^\circ$$

So:

$$\sin150^\circ=\frac12$$ $$\cos150^\circ=-\frac{\sqrt3}{2}$$

Radians and degrees

Full circle:

$$360^\circ=2\pi$$

Half circle:

$$180^\circ=\pi$$

Convert degrees to radians:

$$\boxed{\theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180}}$$

Example:

$$60^\circ=\frac{\pi}{3}$$

Convert radians to degrees:

$$\boxed{\theta_{\text{deg}}=\theta_{\text{rad}}\frac{180}{\pi}}$$

Arc length with radians

If $\theta$ is in radians:

$$\boxed{s=r\theta}$$

This is much simpler than the degree formula.

Example:

$$r=6,\quad\theta=\frac{\pi}{3}$$ $$s=6\cdot\frac{\pi}{3}=2\pi$$

Basic identity

The most important trig identity is:

$$\boxed{\sin^2\theta+\cos^2\theta=1}$$

If:

$$\sin\theta=\frac35$$

and $\theta$ is acute, then:

$$\cos^2\theta=1-\frac9{25}=\frac{16}{25}$$

so:

$$\cos\theta=\frac45$$

Tangent from sine/cosine

$$\boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}$$

If:

$$\sin\theta=\frac{5}{13},\quad \cos\theta=\frac{12}{13}$$

then:

$$\tan\theta=\frac5{12}$$

Solve simple trig equations

Suppose:

$$\sin\theta=\frac12$$

for:

$$0^\circ\le\theta<360^\circ$$

Reference angle:

$$30^\circ$$

Sine positive in Quadrants I and II.

Solutions:

$$\boxed{30^\circ,\ 150^\circ}$$

The interval matters.

Cosine equation

$$\cos\theta=-\frac{\sqrt2}{2}$$

Reference angle:

$$45^\circ$$

Cosine negative in II and III.

So:

$$\boxed{135^\circ,\ 225^\circ}$$

for one full revolution.

Trig graph basics

For:

$$y=\sin x$$

range:

$$[-1,1]$$

period:

$$2\pi$$

For:

$$y=\cos x$$

same range and period.

For:

$$y=\tan x$$

period:

$$\pi$$

and vertical asymptotes occur where cosine is zero.

ACT can combine trig with function transformations.

Amplitude

For:

$$y=A\sin x$$

amplitude:

$$|A|$$

Example:

$$y=-3\sin x$$

amplitude:

$$3$$

negative sign reflects graph.

Period changes

For:

$$y=\sin(Bx)$$

period:

$$\frac{2\pi}{|B|}$$

For tangent:

$$\frac{\pi}{|B|}$$

Know this only if the question clearly enters trig-function graph territory; do not overprepare beyond ACT scope.

Law of sines

For any triangle:

$$\boxed{\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}}$$

Useful when you know:

• an angle-side pair;
• another angle or side.

Example:

$$\frac{a}{\sin30^\circ}=\frac{10}{\sin45^\circ}$$

Solve $a$.

Law of cosines

$$\boxed{c^2=a^2+b^2-2ab\cos C}$$

Generalizes Pythagorean theorem.

If:

$$C=90^\circ$$

then:

$$\cos90^\circ=0$$

and formula becomes:

$$c^2=a^2+b^2$$

Use when:

• two sides and included angle are known;
• or all three sides and an angle is needed.

Common ACT trig traps

Trap 1

Wrong reference angle.

Trap 2

Opposite/adjacent chosen relative to wrong angle.

Trap 3

Calculator in radians when question uses degrees.

Trap 4

Inverse trig confused with reciprocal.

Trap 5

Only one solution given for trig equation over $0^\circ$ to $360^\circ$.

Trap 6

Special-triangle exact value replaced by inaccurate decimal too early.

Calculator mode check

Before a trig question, look at units.

If angle shows:

$$35^\circ$$

calculator must be in degrees.

If angle is:

$$\frac{\pi}{6}$$

radian mode is natural.

A mode error can produce a plausible-looking wrong answer.

Worked example: height

A 20-meter cable forms a $50^\circ$ angle with the ground.

Vertical height:

$$20\sin50^\circ$$

Why sine?

• hypotenuse = 20;
• height = opposite.

Worked example: distance

From a point on the ground, angle of elevation to top of tower is $40^\circ$. Horizontal distance is 30 m.

Height:

$$30\tan40^\circ$$

Worked example: identity

If:

$$\cos\theta=\frac8{17}$$

acute $\theta$:

$$\sin\theta=\frac{15}{17}$$

from 8-15-17 triangle or identity.

Fast ACT trig routine

Right triangle

Label:

• H;
• O;
• A.

Then choose ratio.

Unit circle

Find:

• reference angle;
• quadrant;
• sign.

Equation

Find all solutions in interval.

Word problem

Draw triangle and label units.

Calculator

Check degrees/radians.

Current ACT relevance

Current ACT Math explicitly includes solving missing values in triangles and circles and using trigonometric ratios. ACT's current readiness materials also reference unit-circle trigonometry and basic identities as advanced mathematical progression.

Geometry is roughly 17–20% of ACT Math.

Practice set

1

If opposite $=3$, hypotenuse $=5$:

$$\sin\theta=\boxed{\frac35}$$

2

$$\sin30^\circ=\boxed{\frac12}$$

3

$$\cos120^\circ=\boxed{-\frac12}$$

4

$$180^\circ=\boxed{\pi\text{ radians}}$$

How to review mistakes

Bad note:

trig error.

Better:

I used cosine because 12 was adjacent to the right angle, not adjacent to the reference angle.

Or:

My calculator was in radian mode for a degree question.

Reference angles make unit-circle questions simpler

Instead of memorizing every angle, reduce to a familiar acute reference angle.

Example:

$$210^\circ$$

is:

$$180^\circ+30^\circ$$

Reference angle:

$$30^\circ$$

Quadrant III:

• sine negative;
• cosine negative;
• tangent positive.

So:

$$\sin210^\circ=-\frac12$$ $$\cos210^\circ=-\frac{\sqrt3}{2}$$ $$\tan210^\circ=\frac1{\sqrt3}$$

The reference-angle method compresses the unit circle into a few exact values plus signs.

Coterminal angles

Angles differing by full rotations have the same terminal side.

Degrees:

$$\theta+360^\circ k$$

Radians:

$$\theta+2\pi k$$

Example:

$$-30^\circ$$

is coterminal with:

$$330^\circ$$

This can simplify negative-angle questions.

Solve a tangent equation

$$\tan\theta=1$$

for:

$$0^\circ\le\theta<360^\circ$$

Reference angle:

$$45^\circ$$

Tangent positive in Quadrants I and III.

Solutions:

$$\boxed{45^\circ,\ 225^\circ}$$

Trig identities beyond the basic one

From:

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

you can derive:

$$1+\tan^2\theta=\sec^2\theta$$

if secant/cotangent appear.

ACT usually gives enough context, but recognizing basic identity relationships can shorten work.

Right-triangle inverse problem

A ladder is 15 ft long and reaches 12 ft high.

Find angle with ground.

Adjacent side:

$$\sqrt{15^2-12^2}=9$$

Then:

$$\cos\theta=\frac9{15}=\frac35$$

So:

$$\theta=\cos^{-1}\left(\frac35\right)$$

This shows that Pythagorean theorem and trig often combine.

Law of sines ambiguous case

Law of sines can sometimes produce two possible angles because:

$$\sin\theta=\sin(180^\circ-\theta)$$

ACT questions should provide enough information to identify the valid triangle, but if a sine-based angle seems ambiguous, inspect:

• total angle sum;
• side lengths;
• diagram/context.

Do not blindly accept one inverse-sine output.

Law of cosines as distance formula in disguise

When two sides and included angle are known:

$$c^2=a^2+b^2-2ab\cos C$$

This generalizes the right-triangle distance relationship.

If the included angle is obtuse, cosine is negative, making $c$ larger.

That provides a reasonableness check.

Trigonometric models

A periodic situation can be modeled by sine/cosine:

$$y=A\sin(Bx)+D$$

where:

• amplitude $=|A|$;
• midline $=D$;
• period $=\frac{2\pi}{|B|}$.

ACT may connect trig to Functions rather than present it as a pure triangle question.

Degrees vs radians in formulas

Arc length:

$$s=r\theta$$

requires radians.

If angle is degrees, convert first or use:

$$s=\frac{\theta}{360^\circ}(2\pi r)$$

Mixing degree input into $s=r\theta$ is a common mistake.

Review checklist

When a trig problem is wrong, identify:

• O/A/H labeling;
• reference-angle error;
• quadrant sign;
• degree/radian mode;
• inverse vs reciprocal;
• incomplete solution set;
• special-triangle value;
• law-of-sines/cosines setup.

That converts a vague "trig weakness" into a fixable rule.

Trig equations with a transformed argument

Suppose:

$$\sin(2\theta)=1$$

for:

$$0^\circ\le\theta<180^\circ.$$

Sine equals 1 when:

$$2\theta=90^\circ$$

within the relevant doubled interval.

So:

$$\boxed{\theta=45^\circ}$$

The general habit is:

1. solve for the angle expression;
2. then solve for $\theta$;
3. enforce the original interval.

Use geometry before calculator trig

Suppose a right triangle has sides:

$$7,24,25.$$

If $\theta$ is opposite side 7:

$$\sin\theta=\frac7{25}$$

You may not need $\theta$ at all if the question asks for:

• cosine;
• tangent;
• another side ratio.

Avoid inverse trig unless an angle measure is actually requested.

Similar triangles and trig ratios

If two right triangles share an acute angle, their trig ratios match.

That means you can sometimes solve a trig problem through similarity without a calculator.

Example:

$$\tan\theta=\frac34$$

Any similar triangle with opposite:adjacent $=3:4$ works.

A 6-8-10 triangle therefore represents the same tangent ratio.

Bearings and navigation

A navigation problem may give a direction and distance.

Convert motion into horizontal/vertical components:

$$x=r\cos\theta$$ $$y=r\sin\theta$$

This is vector decomposition using trig.

Example:

A displacement of 20 units at $30^\circ$ above the x-axis:

$$x=20\cos30^\circ=10\sqrt3$$ $$y=20\sin30^\circ=10$$

This links trig with vectors.

Trig reasonableness checks

For an acute angle:

$$0<\sin\theta<1$$ $$0<\cos\theta<1$$

If opposite side is shorter than hypotenuse, the sine cannot exceed 1.

If a calculator gives an impossible ratio, inspect setup or mode.

Review by question family

Classify trig misses into:

• right-triangle ratio;
• special triangle;
• unit circle;
• radians;
• identity;
• equation;
• law of sines/cosines;
• graph/model;
• calculator mode.

Then practice the exact weak family instead of doing random trig questions.

FAQ: ACT trigonometry

Is SOHCAHTOA enough?

It covers right-triangle basics, but harder ACT items can use special triangles, radians, unit-circle values, identities, and non-right-triangle relationships.

Do I need the whole unit circle memorized?

Know the common $30^\circ,45^\circ,60^\circ$ reference-angle values and quadrant signs. That gives you much of the useful structure.

Can ACT test law of sines/cosines?

They are appropriate high-school trigonometry tools and worth knowing for non-right triangles.

What's the biggest calculator trap?

Wrong degree/radian mode.

Related MastaPrep guides

• ACT Geometry and Trigonometry
• ACT Circles and Conic Sections
• ACT Math Formulas
• ACT Functions
• Enhanced ACT 2026–27 Guide

Final takeaway

For right triangles:

SOH-CAH-TOA

For unit-circle questions:

$$\boxed{(\cos\theta,\sin\theta)}$$

For equations:

reference angle + quadrant + interval.

Most ACT trig errors come from choosing the wrong relationship, not from difficult arithmetic.

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Official ACT references:
https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html
https://www.act.org/content/act/en/college-and-career-readiness/benchmarks/ideas-for-progress.html
https://www.act.org/content/dam/act/unsecured/documents/R2519-Design-Framework-for-the-ACT-Enhancements-2026-02.pdf

Trademark note: ACT® is a registered trademark of ACT Education Corp. MastaPrep is independent and is not affiliated with or endorsed by ACT Education Corp.