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SAT
Trigonometry

Master sine, cosine, and tangent ratios, the unit circle, and essential trig identities to conquer every SAT trig question.

SAT Math

What Is Trigonometry?

Trigonometry is the study of relationships between angles and sides in triangles. On the SAT, trig questions focus on right triangles and the unit circle.

Ratios
Sine, cosine, and tangent relate an angle to side lengths.
Unit Circle
Extends trig beyond right triangles to all angles.
Identities
Equations that simplify complex trig expressions.

The SOH-CAH-TOA Ratios

A B C \(\theta\) Adjacent Opposite Hypotenuse
SOH
\(\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}\)
CAH
\(\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}\)
TOA
\(\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}\)

Complementary Angle Relationship

In a right triangle, the two acute angles add to \(90^\circ\). This gives a powerful identity that appears frequently on the SAT:

The Unit Circle

\(45^\circ\) \(\left(\tfrac{\sqrt{2}}{2},\,\tfrac{\sqrt{2}}{2}\right)\) \(\cos\theta\) \(\sin\theta\) x y Q I Q II Q III Q IV

The Pythagorean Identity

Since any point on the unit circle satisfies \(x^2 + y^2 = 1\), substituting \(\cos\theta\) and \(\sin\theta\) gives the most important trig identity:

This means if you know one ratio, you can always find the other:

Find sine
\(\sin\theta = \pm\sqrt{1 - \cos^2\theta}\)
Find cosine
\(\cos\theta = \pm\sqrt{1 - \sin^2\theta}\)

Radians vs Degrees

The SAT uses both degrees and radians. A radian measures the angle by the arc length on a unit circle. The conversion is straightforward:

Degrees Radians
\(0^\circ\)\(0\)
\(30^\circ\)\(\dfrac{\pi}{6}\)
\(45^\circ\)\(\dfrac{\pi}{4}\)
\(60^\circ\)\(\dfrac{\pi}{3}\)
\(90^\circ\)\(\dfrac{\pi}{2}\)
\(180^\circ\)\(\pi\)
\(360^\circ\)\(2\pi\)

Sine vs Cosine: Key Differences

\(\sin\theta\) \(\cos\theta\)
Ratio\(\dfrac{\text{Opposite}}{\text{Hypotenuse}}\)\(\dfrac{\text{Adjacent}}{\text{Hypotenuse}}\)
At \(0^\circ\)01
At \(90^\circ\)10
Unit circle axisy-coordinatex-coordinate
Graph starts atOrigin (0, 0)Peak (0, 1)
Relationship\(\sin\theta = \cos(90^\circ - \theta)\)

How to Solve SAT Trig Problems

1
Identify the triangle or context. Is it a right triangle? A unit circle reference? An identity question?
2
Label everything. Mark the angle, the opposite side, adjacent side, and hypotenuse relative to that angle.
3
Choose the right ratio. Use SOH-CAH-TOA: pick the ratio that connects what you know to what you need.
4
Check for identities. If you see \(\sin^2\theta + \cos^2\theta\), complementary angles, or \(\tan\theta = \sin\theta/\cos\theta\), simplify first.
5
Verify with known values. Plug in special angles (\(30^\circ\), \(45^\circ\), \(60^\circ\)) to check your answer makes sense.

Example 1: Finding a Side

A B C \(35^\circ\) 10 x

Problem: Find side \(x\) (opposite to the \(35^\circ\) angle) when the hypotenuse is 10.

Example 2: Using an Identity

Problem: If \(\sin\theta = \dfrac{3}{5}\) and \(\theta\) is in Quadrant I, find \(\cos\theta\).

1
Apply the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)
2
Substitute: \(\left(\dfrac{3}{5}\right)^2 + \cos^2\theta = 1\)
3
Simplify: \(\dfrac{9}{25} + \cos^2\theta = 1 \implies \cos^2\theta = \dfrac{16}{25}\)
4
Solve: \(\cos\theta = +\dfrac{4}{5}\) (positive because Q I)

Example 3: Complementary Angles

Problem: If \(\sin((4x + 10)^\circ) = \cos((2x + 20)^\circ)\), find \(x\).

1
Apply the rule: \(\sin(A) = \cos(B)\) means \(A + B = 90^\circ\).
2
Set up the equation: \((4x + 10) + (2x + 20) = 90\)
3
Combine and solve: \(6x + 30 = 90 \implies 6x = 60 \implies x = 10\)

Understanding Tangent

Tangent is the third core ratio. It connects directly to sine and cosine:

Undefined at \(90^\circ\)
When \(\cos\theta = 0\), tangent is undefined. This happens at \(90^\circ\) and \(270^\circ\).
Sign by quadrant
Tangent is positive in Q I and Q III (where sine and cosine share the same sign).

Special Right Triangles

\(45^\circ\) \(45^\circ\) 1 1 \(\sqrt{2}\) 45-45-90 \(30^\circ\) \(60^\circ\) \(\sqrt{3}\) 1 2 30-60-90
Angle \(\sin\) \(\cos\) \(\tan\)
\(30^\circ\)\(\frac{1}{2}\)\(\frac{\sqrt{3}}{2}\)\(\frac{1}{\sqrt{3}}\)
\(45^\circ\)\(\frac{\sqrt{2}}{2}\)\(\frac{\sqrt{2}}{2}\)\(1\)
\(60^\circ\)\(\frac{\sqrt{3}}{2}\)\(\frac{1}{2}\)\(\sqrt{3}\)

Common Mistakes

Mixing up sides
Using the adjacent side when the problem asks for the opposite. Always label sides relative to the given angle.
Calculator in wrong mode
Computing \(\sin(\pi/6)\) in degree mode gives a wrong answer. Check if the problem uses radians or degrees.
Forgetting quadrant signs
In Q II, sine is positive but cosine is negative. Use the mnemonic: "All Students Take Calculus" for sign rules.

SAT Trig Cheat Sheet

Concept Formula / Rule When to Use
SOH\(\sin\theta = \frac{\text{Opp}}{\text{Hyp}}\)Know hyp, need opp (or vice versa)
CAH\(\cos\theta = \frac{\text{Adj}}{\text{Hyp}}\)Know hyp, need adj (or vice versa)
TOA\(\tan\theta = \frac{\text{Opp}}{\text{Adj}}\)Know adj, need opp (or vice versa)
Pythagorean\(\sin^2\theta + \cos^2\theta = 1\)Know \(\sin\theta\) or \(\cos\theta\), find the other
Complementary\(\sin x = \cos(90^\circ - x)\)See \(\sin A = \cos B\) equation
Tangent\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)Convert between ratios

Question 1 of 5

In a right triangle, the side opposite angle \(A\) is 5 and the hypotenuse is 13. What is \(\sin A\)?

Question 2 of 5

If \(\sin((3x + 15)^\circ) = \cos((x + 25)^\circ)\), what is the value of \(x\)?

Question 3 of 5

If \(\cos\theta = \dfrac{7}{25}\) and \(\theta\) is in Quadrant I, what is \(\sin\theta\)?

Question 4 of 5

What is the value of \(\tan 45^\circ\)?

Question 5 of 5

Which expression is equivalent to \(\dfrac{\sin\theta}{\cos\theta}\)?