Does the ACT give me the trig formulas, or do I have to memorize them?

You should memorize them. The ACT does not provide a formula sheet, so know SOHCAHTOA, the Pythagorean identity, the degree-radian conversion, and both the law of sines and law of cosines going in.

How much trigonometry is actually on the ACT?

Trig typically makes up a small but consistent portion of the Math section, roughly four to six questions. Most are right-triangle SOHCAHTOA problems; a few involve the unit circle, radians, identities, or the laws of sines and cosines.

Should my calculator be in degrees or radians?

Match the mode to the problem. If angles are given in degrees, use degree mode; if they use or radian measure, switch to radian mode. Setting the wrong mode is one of the most common avoidable errors.

When do I use the law of sines versus the law of cosines?

Use the law of cosines when you know two sides and the included angle, or all three sides. Use the law of sines when you have an angle paired with its opposite side plus one additional angle or side.

ACT-2.9

Trigonometry

Master ACT trigonometry: SOHCAHTOA, the unit circle, radians, the Pythagorean identity, and the laws of sines and cosines to nail every angle and side-length question.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Trigonometry, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Trigonometry free

What this lesson covers

Trigonometry questions show up on nearly every ACT Math section, usually four to six of them, and they range from a quick SOHCAHTOA setup to a multi-step problem using the law of cosines. The good news: the ACT tests a predictable core of skills. If you can label a right triangle correctly, read the unit circle, convert between degrees and radians, and recognize when a triangle is not right-angled, you can handle almost anything they throw at you.

This lesson builds that toolkit from the ground up. You will learn the three primary ratios, the reciprocal functions, the identity that links sine and cosine, and the two laws that extend trigonometry to any triangle. Along the way we point out the traps the ACT loves to set.

SOHCAHTOA and the Right Triangle

Every right-triangle question starts by identifying three sides relative to a chosen angle

\theta

: the opposite side (across from

\theta

), the adjacent side (next to

\theta

, not the hypotenuse), and the hypotenuse (across from the right angle). The mnemonic SOHCAHTOA encodes the three primary ratios.

Function	Ratio	Memory
$\sin\theta$	$\frac{\text{opp}}{\text{hyp}}$	SOH
$\cos\theta$	$\frac{\text{adj}}{\text{hyp}}$	CAH
$\tan\theta$	$\frac{\text{opp}}{\text{adj}}$	TOA

The reciprocal functions appear less often but are fair game:

\csc\theta=\frac{1}{\sin\theta}

\sec\theta=\frac{1}{\cos\theta}

, and

\cot\theta=\frac{1}{\tan\theta}

.

A classic trap: the ACT gives you a triangle where the side you want is neither opposite nor adjacent in the obvious way, or it labels the angle at the top. Always redraw and re-label from the specific angle in the question. Another trap is mixing up which ratio to use — pick the one that connects the side you know with the side you want. To find an unknown angle from two sides, use inverse functions like

\theta=\tan^{-1}\!\left(\frac{\text{opp}}{\text{adj}}\right)

The Unit Circle and Radian Measure

The unit circle is a circle of radius

1

centered at the origin. For an angle

\theta

measured counterclockwise from the positive

x

-axis, the point on the circle is

(\cos\theta,\sin\theta)

. This is why

\cos

is the

x

-coordinate and

\sin

is the

y

-coordinate — a fact the ACT tests directly.

Radians measure angles by arc length. A full circle is

2\pi

radians

=360^\circ

, so the conversion is

180^\circ=\pi

radians. To convert, multiply degrees by

\frac{\pi}{180}

, or multiply radians by

\frac{180}{\pi}

Degrees	Radians	$\sin$	$\cos$
$0^\circ$	$0$	$0$	$1$
$30^\circ$	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt3}{2}$
$45^\circ$	$\frac{\pi}{4}$	$\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$
$60^\circ$	$\frac{\pi}{3}$	$\frac{\sqrt3}{2}$	$\frac{1}{2}$
$90^\circ$	$\frac{\pi}{2}$	$1$	$0$

Signs depend on the quadrant: in Quadrant II sine is positive and cosine negative; in Quadrant III both are negative; in Quadrant IV cosine is positive and sine negative. A common mistake is leaving your calculator in the wrong mode — if a question uses

\pi

, switch to radians.

The Pythagorean Identity and Basic Manipulation

Because the point

(\cos\theta,\sin\theta)

lies on a circle of radius

1

, the Pythagorean theorem gives the single most tested identity:

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

This lets you find one ratio from another. If

\sin\theta=\frac{3}{5}

, then

\cos^2\theta=1-\frac{9}{25}=\frac{16}{25}

, so

\cos\theta=\pm\frac{4}{5}

; the sign depends on the quadrant. Dividing the identity through by

\cos^2\theta

produces

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

, which occasionally appears.

Also remember the quotient relationship

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

. Many ACT problems that look intimidating collapse quickly once you rewrite tangent this way or apply the identity.

A frequent misconception is writing

\sin^2\theta

\sin\theta^2

; the exponent applies to the whole function value, meaning

(\sin\theta)^2

. Another is forgetting the

\pm

sign when taking a square root — the ACT often includes a distractor with the wrong sign to catch students who ignore the quadrant information given in the problem.

Laws of Sines and Cosines for Any Triangle

When a triangle has no right angle, SOHCAHTOA does not apply. Two laws extend trigonometry to all triangles, where sides

a

b

c

are opposite angles

A

B

C

.

The law of sines relates each side to the sine of its opposite angle:

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

Use it when you know an angle and its opposite side plus one more piece (angle-angle-side or angle-side-angle, and some side-side-angle cases).

The law of cosines generalizes the Pythagorean theorem:

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

Use it when you know two sides and the included angle (to find the third side), or all three sides (to find an angle). Notice that if

C=90^\circ

\cos C=0

and the formula reduces to

c^2=a^2+b^2

You know	Use
Angle + opposite side + one more	Law of sines
Two sides + included angle	Law of cosines
All three sides	Law of cosines

The ACT usually gives you these formulas indirectly or expects recall, so memorize both. Match the given information to the correct law before plugging in.

Key terms

SOHCAHTOA.: A mnemonic giving the three primary right-triangle ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
Unit circle.: A circle of radius $1$ centered at the origin; the point at angle $\theta$ has coordinates $(\cos\theta,\sin\theta)$ .
Radian.: An angle measure based on arc length, where $\pi$ radians equals $180^\circ$ and a full circle is $2\pi$ radians.
Pythagorean identity.: The relationship $\sin^2\theta+\cos^2\theta=1$ , true for every angle $\theta$ .
Law of sines.: For any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ , linking each side to the sine of the opposite angle.
Law of cosines.: For any triangle, $c^2=a^2+b^2-2ab\cos C$ , a generalization of the Pythagorean theorem to non-right triangles.
Inverse trig function.: An operation such as $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ that returns the angle whose ratio you supply.
Reciprocal functions.: Cosecant, secant, and cotangent, defined as $\csc\theta=\frac{1}{\sin\theta}$ , $\sec\theta=\frac{1}{\cos\theta}$ , and $\cot\theta=\frac{1}{\tan\theta}$ .

Worked example

In right triangle

ABC

, the right angle is at

C

. The angle at

A

\theta

, the side opposite

A

has length

7

, and the hypotenuse has length

25

. Find

\cos\theta

and the length of the side adjacent to

\theta

Start by identifying sides relative to angle

\theta

at vertex

A

. The side opposite

\theta

7

, and the hypotenuse (opposite the right angle at

C

) is

25

.

First find the adjacent side with the Pythagorean theorem. If the adjacent side is

b

, then

7^2+b^2=25^2

, so

49+b^2=625

, giving

b^2=576

and

b=24

.

Now apply CAH:

\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}

.

As a check, use the Pythagorean identity. Here

\sin\theta=\frac{7}{25}

, so

\sin^2\theta+\cos^2\theta=\frac{49}{625}+\frac{576}{625}=\frac{625}{625}=1

. The identity holds, confirming our answer.

So

\cos\theta=\frac{24}{25}

and the adjacent side is

24

Practice questions

A right triangle has an acute angle

\theta

with

\sin\theta=\frac{5}{13}

. What is the value of

\tan\theta

$\frac{5}{12}$
$\frac{12}{13}$
$\frac{12}{5}$
$\frac{13}{5}$

Answer: $\frac{5}{12}$

Since

\sin\theta=\frac{5}{13}

, the opposite side is

5

and the hypotenuse is

13

. Find the adjacent side:

\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12

. Then

\tan\theta=\frac{\text{opp}}{\text{adj}}=\frac{5}{12}

. The choice

\frac{12}{5}

is the reciprocal (cotangent), a common trap.

A triangle has sides of length

8

and

11

with an included angle of

40^\circ

between them. Explain which law you would use and set up the expression for the length of the third side.

Answer: Use the law of cosines: the third side $c$ satisfies $c^2=8^2+11^2-2(8)(11)\cos 40^\circ$ .

You are given two sides and the angle between them (SAS), which is exactly the situation the law of cosines handles. Substitute:

c^2=64+121-176\cos 40^\circ

. Evaluating,

\cos 40^\circ\approx 0.766

, so

c^2\approx 185-134.8=50.2

, giving

c\approx 7.1

. The law of sines would not work first here because no angle is paired with its opposite side.

Convert

135^\circ

to radians and state the sign of

\cos 135^\circ

Answer: $135^\circ=\frac{3\pi}{4}$ radians, and $\cos 135^\circ$ is negative.

Multiply by

\frac{\pi}{180}

135\cdot\frac{\pi}{180}=\frac{135\pi}{180}=\frac{3\pi}{4}

. This angle lies in Quadrant II, where the

x

-coordinate on the unit circle is negative, so cosine is negative. In fact

\cos 135^\circ=-\frac{\sqrt2}{2}

FAQ

Does the ACT give me the trig formulas, or do I have to memorize them?: You should memorize them. The ACT does not provide a formula sheet, so know SOHCAHTOA, the Pythagorean identity, the degree-radian conversion, and both the law of sines and law of cosines going in.
How much trigonometry is actually on the ACT?: Trig typically makes up a small but consistent portion of the Math section, roughly four to six questions. Most are right-triangle SOHCAHTOA problems; a few involve the unit circle, radians, identities, or the laws of sines and cosines.
Should my calculator be in degrees or radians?: Match the mode to the problem. If angles are given in degrees, use degree mode; if they use $\pi$ or radian measure, switch to radian mode. Setting the wrong mode is one of the most common avoidable errors.
When do I use the law of sines versus the law of cosines?: Use the law of cosines when you know two sides and the included angle, or all three sides. Use the law of sines when you have an angle paired with its opposite side plus one additional angle or side.

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