Trigonometry Basics Study Guide

This trigonometry basics study guide explains the core ideas students need before moving into advanced trig, precalculus, or calculus. You will review how sine, cosine, and tangent connect angles to side lengths in right triangles, how the unit circle extends trig functions beyond acute angles, and how common identities help simplify expressions and solve equations. Use the notes, flashcards, and quiz below to strengthen your understanding and prepare for homework, tests, or a cumulative math review.

Key takeaways

• Trigonometry studies relationships between angles and side lengths, especially in right triangles and on the unit circle.
• The three primary trig ratios are sine, cosine, and tangent: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.
• Special right triangles, such as 30-60-90 and 45-45-90 triangles, help you find exact trig values quickly.
• The unit circle defines trig functions for all angle measures, not just angles inside right triangles.
• Key identities, including sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ, are essential for simplifying trig expressions and solving equations.

What Is Trigonometry?

Trigonometry is the branch of math that studies the relationship between angles and side lengths. In basic trig, most problems begin with a right triangle. A right triangle has one 90° angle, and the side across from that angle is called the hypotenuse. The other two sides are described relative to a chosen angle: the side across from the angle is the opposite side, and the side next to the angle is the adjacent side. Trigonometry lets you use one angle and one side, or multiple side lengths, to find missing measurements.

The Three Main Trig Ratios: Sine, Cosine, and Tangent

The most important trigonometry notes start with SOH-CAH-TOA. This memory tool helps you remember the main ratios: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Written as formulas: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. For example, if an angle has an opposite side of 3 and a hypotenuse of 5, then sin θ = 3/5. These ratios are used to solve for unknown sides and angles in right triangles.

Special Right Triangles and Exact Values

Two special triangles appear often in trigonometry basics: the 45-45-90 triangle and the 30-60-90 triangle. A 45-45-90 triangle has side ratios 1 : 1 : √2, meaning the legs are equal and the hypotenuse is √2 times a leg. A 30-60-90 triangle has side ratios 1 : √3 : 2, where the side across 30° is the shortest side, the side across 60° is √3 times that side, and the hypotenuse is twice the shortest side. These triangles help you memorize exact values such as sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, and sin 45° = √2/2.

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin on the coordinate plane. It expands trigonometry beyond right triangles by defining trig values for any angle. For an angle θ measured from the positive x-axis, the point on the unit circle has coordinates (cos θ, sin θ). This means cosine is the x-coordinate and sine is the y-coordinate. Common angles include 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°. The unit circle is also useful for understanding positive and negative trig values in different quadrants.

Degrees, Radians, and Angle Measurement

Angles in trigonometry can be measured in degrees or radians. A full circle is 360° or 2π radians, so 180° equals π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Common conversions include 30° = π/6, 45° = π/4, 60° = π/3, and 90° = π/2. Understanding both systems is important because many higher-level math courses use radians more often than degrees.

Basic Trigonometric Identities

A trigonometric identity is an equation that is true for all values where both sides are defined. The most important identity for beginners is the Pythagorean identity: sin²θ + cos²θ = 1. Another key identity is tan θ = sin θ/cos θ. Reciprocal identities also appear often: csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. These identities are used to simplify expressions, prove relationships, and solve trig equations.

Flashcards

Quiz

1. Which ratio correctly defines sine in a right triangle?

1. A. opposite/hypotenuse
2. B. adjacent/hypotenuse
3. C. opposite/adjacent
4. D. hypotenuse/opposite

2. Which ratio correctly defines cosine in a right triangle?

1. A. opposite/hypotenuse
2. B. adjacent/hypotenuse
3. C. opposite/adjacent
4. D. hypotenuse/adjacent

3. What are the coordinates of a point on the unit circle at angle θ?

1. A. (sin θ, cos θ)
2. B. (tan θ, sin θ)
3. C. (cos θ, sin θ)
4. D. (cos θ, tan θ)

4. Which identity is the basic Pythagorean identity?

1. A. sin θ + cos θ = 1
2. B. sin²θ + cos²θ = 1
3. C. tan²θ + cos²θ = 1
4. D. sin²θ – cos²θ = 1

5. What is the value of tan 45°?

1. A. 0
2. B. 1/2
3. C. 1
4. D. √3

6. Which angle measure is equivalent to π radians?

1. A. 90°
2. B. 180°
3. C. 270°
4. D. 360°

7. In a 30-60-90 triangle, what is the ratio of the sides?

1. A. 1 : 1 : √2
2. B. 1 : √3 : 2
3. C. 1 : 2 : √3
4. D. √2 : √2 : 1

FAQs

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