# Trigonometry Review Exercises With Solutions

## Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental part of mathematics and is used in various fields, including engineering, science, and architecture. In this article, we will provide you with tips and exercises to help you review and improve your trigonometry skills.

## Basics of Trigonometry

Before we dive into the exercises, let's have a quick review of the basics of trigonometry. Trigonometry has six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These functions are used to relate the angles of a triangle to its sides. The sine function is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

### Exercise 1

Calculate the sine, cosine, and tangent of a 30-degree angle.

Solution: The sine of 30 degrees is 0.5, cosine is 0.87, and tangent is 0.58.

### Exercise 2

Find the value of x in a right-angled triangle with a hypotenuse of 10 and an angle of 30 degrees.

Solution: Using the sine function, we can determine that x equals 5.

## Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables. They are useful in simplifying trigonometric expressions and solving trigonometric equations. Some of the commonly used trigonometric identities are sine squared plus cosine squared equals one, tangent equals sine over cosine, and secant equals one over cosine.

### Exercise 3

Simplify the expression: cos(x) sec(x).

Solution: Using the identity secant equals one over cosine, we can simplify the expression to one.

### Exercise 4

Prove the identity: sine(x) over cosine(x) equals tangent(x).

Solution: Using the definition of tangent as opposite over adjacent and sine as opposite over hypotenuse, we can simplify the expression to opposite over adjacent, which is the definition of tangent.

## Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions. They can be solved using algebraic methods or graphical methods. Some of the commonly used techniques for solving trigonometric equations are factoring, substitution, and using trigonometric identities.

### Exercise 5

Solve the equation: sin(x) equals 0.5.

Solution: Using the inverse sine function, we can determine that x equals 30 degrees or pi over 6 radians.

### Exercise 6

Solve the equation: cos(2x) equals 0.5.

Solution: Using the double-angle formula for cosine, we can simplify the equation to 2cos squared(x) minus 1 equals 0. Solving for cos(x), we get plus or minus square root of 2 over 2. Therefore, x equals pi over 4 or 3pi over 4 radians.

## Conclusion

Trigonometry is an essential part of mathematics that has practical applications in various fields. Reviewing and improving your trigonometry skills can enhance your problem-solving abilities and help you excel in your academic or professional pursuits. We hope that these tips and exercises have been helpful in your trigonometry review.

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