Trigonometry

Trigonometry
Trig

The study of triangles, with emphasis on calculations involving the lengths of sides and the measures of angles. Trigonometry is also known as trig.

Trigonometry is based on the six functions sine, cosine, tangent, cosecant, secant, and cotangent. Trig also includes studies of the properties of these functions and their graphs.

Finally, trig involves the analysis of circles and periodic motion.

Key Formula

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Where:

$\theta$ = An acute angle in a right triangle
$\text{opposite}$ = The side across from the angle θ
$\text{adjacent}$ = The side next to the angle θ (not the hypotenuse)
$\text{hypotenuse}$ = The longest side, opposite the right angle

Worked Example

Problem: A right triangle has an angle of 30° and a hypotenuse of 10. Find the lengths of the other two sides.

Step 1: Identify the known values. The angle θ = 30°, the hypotenuse = 10, and you need the opposite and adjacent sides.

Step 2: Use the sine ratio to find the side opposite the 30° angle.

\sin 30° = \frac{\text{opposite}}{10} \implies \text{opposite} = 10 \cdot \sin 30° = 10 \cdot 0.5 = 5

Step 3: Use the cosine ratio to find the side adjacent to the 30° angle.

\cos 30° = \frac{\text{adjacent}}{10} \implies \text{adjacent} = 10 \cdot \cos 30° = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}

Step 4: Verify with the Pythagorean theorem.

5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 = 10^2 \checkmark

Answer: The opposite side is 5 and the adjacent side is 5√3 ≈ 8.66.

Another Example

This example applies trigonometry to a real-world scenario where you solve for the hypotenuse rather than a leg, showing how to rearrange the trig ratio.

Problem: A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the ladder makes a 65° angle with the ground. How long is the ladder?

Step 1: Draw the situation. The ground, wall, and ladder form a right triangle. The 65° angle is between the ground and the ladder. The 6-foot distance is the side adjacent to this angle, and the ladder is the hypotenuse.

Step 2: Choose the correct trig ratio. You know the adjacent side and need the hypotenuse, so use cosine.

\cos 65° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{L}

Step 3: Solve for L, the length of the ladder.

L = \frac{6}{\cos 65°} = \frac{6}{0.4226} \approx 14.2

Answer: The ladder is approximately 14.2 feet long.

Frequently Asked Questions

What is SOHCAHTOA and how does it help with trigonometry?

SOHCAHTOA is a mnemonic that stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It helps you remember which trig ratio to use in a right triangle problem. Just identify which two sides are involved relative to your angle, and the mnemonic tells you which function connects them.

What is the difference between trigonometry and geometry?

Geometry is the broader study of shapes, sizes, and spatial properties. Trigonometry is a specific branch that focuses on the relationships between angles and side lengths, primarily in triangles. Geometry might ask you to prove that two triangles are similar; trigonometry gives you the tools to calculate exact side lengths and angle measures using sine, cosine, and tangent.

When do you use trigonometry in real life?

Trigonometry is used in engineering, architecture, physics, navigation, and computer graphics. Surveyors use it to measure distances to inaccessible points. Physicists use it to break forces into components. Sound and light waves are modeled with trigonometric functions, making trig essential in music production and optics.

Trigonometry vs. Geometry

	Trigonometry	Geometry
Focus	Relationships between angles and side lengths	Properties of shapes, size, and space
Key tools	Six trig functions (sin, cos, tan, csc, sec, cot)	Postulates, theorems, proofs, constructions
Typical problem	Find a missing side or angle using a trig ratio	Prove two triangles are congruent or find an area
Extends to	Periodic functions, waves, unit circle	3D solids, coordinate geometry, transformations

Why It Matters

Trigonometry appears throughout high school math, from Algebra 2 through pre-calculus, and is essential for success in calculus where trig functions and identities appear constantly. It is also a core topic on standardized tests like the SAT, ACT, and AP exams. Beyond school, trig provides the mathematical language for describing anything that oscillates or rotates—sound waves, alternating current, satellite orbits, and more.

Common Mistakes

Mistake: Using the wrong trig ratio—for example, applying sine when you should use cosine because you confused which side is opposite and which is adjacent.

Correction: Always label the sides relative to the specific angle you are working with. The opposite side is directly across from your angle; the adjacent side is next to your angle and is not the hypotenuse. Then match to SOHCAHTOA.

Mistake: Forgetting to check whether your calculator is set to degrees or radians, leading to completely wrong answers.

Correction: Before computing, verify your calculator mode. If the problem gives angles in degrees (like 30°, 45°, 60°), set it to degree mode. If angles are in radians (like π/6, π/4), use radian mode. A quick check: sin 30° should give 0.5.

Related Terms

SOHCAHTOA — Mnemonic for the three basic trig ratios
Sine — Fundamental trig function: opposite over hypotenuse
Cosine — Fundamental trig function: adjacent over hypotenuse
Tangent — Fundamental trig function: opposite over adjacent
Trig Identities — Equations that are true for all angle values
Unit Circle Trig Definitions — Extends trig beyond right triangles using a circle
Law of Sines — Relates sides and angles in any triangle
Law of Cosines — Generalizes the Pythagorean theorem to any triangle