Simplifying Trigonometric Expressions

Oct 13, 2025 by miodeya.com 96 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will simplify four trigonometric expressions without the use of a calculator.

Simplifying Trigonometric Expressions

a) Simplifying $\sin 75^{\circ}$

To simplify $\sin 75^{\circ}$ , we can use the angle addition formula, which states that $\sin (A + B) = \sin A \cos B + \cos A \sin B$ . We can rewrite $75^{\circ}$ as the sum of $60^{\circ}$ and $15^{\circ}$ .

$\sin 75^{\circ} = \sin (60^{\circ} + 15^{\circ})$

Using the angle addition formula, we get:

$\sin 75^{\circ} = \sin 60^{\circ} \cos 15^{\circ} + \cos 60^{\circ} \sin 15^{\circ}$

We know that $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ , $\cos 60^{\circ} = \frac{1}{2}$ , and $\sin 15^{\circ} = \sin 15^{\circ}$ . Substituting these values, we get:

$\sin 75^{\circ} = \frac{\sqrt{3}}{2} \cos 15^{\circ} + \frac{1}{2} \sin 15^{\circ}$

b) Simplifying $\cos 15^{\circ}$

To simplify $\cos 15^{\circ}$ , we can use the angle addition formula, which states that $\cos (A + B) = \cos A \cos B - \sin A \sin B$ . We can rewrite $15^{\circ}$ as the difference between $30^{\circ}$ and $15^{\circ}$ .

$\cos 15^{\circ} = \cos (30^{\circ} - 15^{\circ})$

Using the angle addition formula, we get:

$\cos 15^{\circ} = \cos 30^{\circ} \cos 15^{\circ} - \sin 30^{\circ} \sin 15^{\circ}$

We know that $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ , $\sin 30^{\circ} = \frac{1}{2}$ , and $\cos 15^{\circ} = \cos 15^{\circ}$ . Substituting these values, we get:

$\cos 15^{\circ} = \frac{\sqrt{3}}{2} \cos 15^{\circ} - \frac{1}{2} \sin 15^{\circ}$

c) Simplifying $\cos(325^{\circ}) \cos(-10^{\circ}) + \sin(215^{\circ}) \sin(10^{\circ})$

To simplify $\cos(325^{\circ}) \cos(-10^{\circ}) + \sin(215^{\circ}) \sin(10^{\circ})$ , we can use the product-to-sum formula, which states that $\cos A \cos B + \sin A \sin B = \cos (A - B)$ . We can rewrite $325^{\circ}$ as the sum of $315^{\circ}$ and $10^{\circ}$ , and $215^{\circ}$ as the sum of $210^{\circ}$ and $5^{\circ}$ .

$\cos(325^{\circ}) \cos(-10^{\circ}) + \sin(215^{\circ}) \sin(10^{\circ}) = \cos (315^{\circ} + 10^{\circ}) \cos (-10^{\circ}) + \sin (210^{\circ} + 5^{\circ}) \sin (10^{\circ})$

Using the product-to-sum formula, we get:

$\cos(325^{\circ}) \cos(-10^{\circ}) + \sin(215^{\circ}) \sin(10^{\circ}) = \cos (315^{\circ}) \cos (-10^{\circ}) - \sin (315^{\circ}) \sin (-10^{\circ}) + \cos (210^{\circ}) \sin (10^{\circ}) + \sin (210^{\circ}) \cos (10^{\circ})$

We know that $\cos 315^{\circ} = -\frac{\sqrt{2}}{2}$ , $\sin 315^{\circ} = \frac{\sqrt{2}}{2}$ , $\cos 210^{\circ} = -\frac{\sqrt{3}}{2}$ , and $\sin 210^{\circ} = \frac{1}{2}$ . Substituting these values, we get:

$\cos(325^{\circ}) \cos(-10^{\circ}) + \sin(215^{\circ}) \sin(10^{\circ}) = -\frac{\sqrt{2}}{2} \cos (-10^{\circ}) + \frac{\sqrt{2}}{2} \sin (-10^{\circ}) - \frac{\sqrt{3}}{2} \sin (10^{\circ}) + \frac{1}{2} \cos (10^{\circ})$

d) Simplifying $\sin(331^{\circ})$

To simplify $\sin(331^{\circ})$ , we can use the angle addition formula, which states that $\sin (A + B) = \sin A \cos B + \cos A \sin B$ . We can rewrite $331^{\circ}$ as the sum of $330^{\circ}$ and $1^{\circ}$ .

$\sin(331^{\circ}) = \sin (330^{\circ} + 1^{\circ})$

Using the angle addition formula, we get:

$\sin(331^{\circ}) = \sin 330^{\circ} \cos 1^{\circ} + \cos 330^{\circ} \sin 1^{\circ}$

We know that $\sin 330^{\circ} = -\frac{\sqrt{3}}{2}$ , $\cos 330^{\circ} = \frac{1}{2}$ , and $\sin 1^{\circ} = \sin 1^{\circ}$ . Substituting these values, we get:

$\sin(331^{\circ}) = -\frac{\sqrt{3}}{2} \cos 1^{\circ} + \frac{1}{2} \sin 1^{\circ}$

Conclusion

Q: What are some common trigonometric identities that can be used to simplify expressions?

A: Some common trigonometric identities that can be used to simplify expressions include:

Angle Addition Formula: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
Angle Subtraction Formula: $\sin (A - B) = \sin A \cos B - \cos A \sin B$
Product-to-Sum Formula: $\cos A \cos B + \sin A \sin B = \cos (A - B)$
Sum-to-Product Formula: $\sin A \cos B + \cos A \sin B = \sin (A + B)$

Q: How can I use the angle addition formula to simplify an expression?

A: To use the angle addition formula to simplify an expression, you can follow these steps:

Identify the angles in the expression that you want to simplify.
Rewrite the expression as the sum or difference of two angles.
Apply the angle addition formula to simplify the expression.

For example, to simplify $\sin (75^{\circ})$ , you can rewrite it as $\sin (60^{\circ} + 15^{\circ})$ and apply the angle addition formula to get $\sin 75^{\circ} = \sin 60^{\circ} \cos 15^{\circ} + \cos 60^{\circ} \sin 15^{\circ}$ .

Q: How can I use the product-to-sum formula to simplify an expression?

A: To use the product-to-sum formula to simplify an expression, you can follow these steps:

Identify the product of two cosine or sine functions in the expression that you want to simplify.
Apply the product-to-sum formula to simplify the expression.

For example, to simplify $\cos (325^{\circ}) \cos (-10^{\circ}) + \sin (215^{\circ}) \sin (10^{\circ})$ , you can apply the product-to-sum formula to get $\cos (325^{\circ}) \cos (-10^{\circ}) + \sin (215^{\circ}) \sin (10^{\circ}) = \cos (315^{\circ} + 10^{\circ}) \cos (-10^{\circ}) + \sin (210^{\circ} + 5^{\circ}) \sin (10^{\circ})$ .

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

Not using the correct trigonometric identities: Make sure to use the correct trigonometric identities to simplify the expression.
Not rewriting the expression in the correct form: Make sure to rewrite the expression in the correct form before applying the trigonometric identities.
Not simplifying the expression completely: Make sure to simplify the expression completely before giving the final answer.

Q: How can I practice simplifying trigonometric expressions?

A: You can practice simplifying trigonometric expressions by:

Working through practice problems: Work through practice problems to get a feel for how to simplify trigonometric expressions.
Using online resources: Use online resources, such as video tutorials and practice problems, to help you learn how to simplify trigonometric expressions.
Seeking help from a teacher or tutor: Seek help from a teacher or tutor if you are having trouble simplifying trigonometric expressions.

Q: What are some real-world applications of simplifying trigonometric expressions?

A: Simplifying trigonometric expressions has many real-world applications, including:

Navigation: Simplifying trigonometric expressions is used in navigation to calculate distances and angles between two points.
Physics: Simplifying trigonometric expressions is used in physics to calculate the motion of objects and the forces acting on them.
Engineering: Simplifying trigonometric expressions is used in engineering to design and optimize systems and structures.

By mastering the art of simplifying trigonometric expressions, you can develop a deeper understanding of the subject and apply it to real-world problems.