Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Problem 5.34

Textbook Question

Simplify each expression.
√[(1 + cos 76°)/2]

Verified step by step guidance

Recognize that the expression \( \sqrt{\frac{1 + \cos 76^\circ}{2}} \) is related to a trigonometric identity.

Recall the half-angle identity for cosine: \( \cos \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 + \cos \theta}{2}} \).

Identify that \( \theta = 76^\circ \) in the given expression, so we are looking for \( \cos \left( \frac{76^\circ}{2} \right) \).

Calculate \( \frac{76^\circ}{2} \) to find the angle for which we need the cosine value.

Express the simplified form as \( \cos \left( 38^\circ \right) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In the context of the expression given, cos 76° represents the cosine of 76 degrees, which is a specific value that can be calculated or approximated using a calculator or trigonometric tables.

Half-Angle Identity

The half-angle identities are formulas that express trigonometric functions of half angles in terms of the functions of the original angle. For example, the identity for cosine states that cos(θ/2) = √[(1 + cos θ)/2]. This identity is particularly useful for simplifying expressions involving angles that are halved, such as the expression in the question.

Square Root Function

The square root function is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In the expression √[(1 + cos 76°)/2], the square root is applied to the result of the fraction, which simplifies the expression further. Understanding how to manipulate square roots is essential for simplifying trigonometric expressions effectively.