Understanding the Trigonometric Expression

Oct 13, 2025 by windailys.com 92 views

If $\theta = 30^{\circ}$ , Evaluate the Trigonometric Expression without Using Tables

In this article, we will explore the evaluation of a trigonometric expression involving the cosine function. The expression to be evaluated is $\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)}$ , where $\theta = 30^{\circ}$ . We will use trigonometric identities and formulas to simplify the expression and find its value.

Understanding the Trigonometric Expression

The given expression involves the cosine function and its arguments are in the form of $45^{\circ} + \theta$ and $45^{\circ} - \theta$ . To evaluate this expression, we need to use trigonometric identities that relate the cosine function to other trigonometric functions.

Using the Angle Addition Formula

The angle addition formula for cosine states that:

\cos (A + B) = \cos A \cos B - \sin A \sin B

We can use this formula to expand the expression $\cos (45^{\circ} + \theta)$ .

\cos (45^{\circ} + \theta) = \cos 45^{\circ} \cos \theta - \sin 45^{\circ} \sin \theta

Similarly, we can use the angle subtraction formula for cosine to expand the expression $\cos (45^{\circ} - \theta)$ .

\cos (45^{\circ} - \theta) = \cos 45^{\circ} \cos \theta + \sin 45^{\circ} \sin \theta

Simplifying the Expression

Now that we have expanded the expressions using the angle addition and subtraction formulas, we can simplify the given expression.

\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)} = \frac{\cos 45^{\circ} \cos \theta - \sin 45^{\circ} \sin \theta}{\cos 45^{\circ} \cos \theta + \sin 45^{\circ} \sin \theta}

We can simplify this expression further by dividing both the numerator and denominator by $\cos 45^{\circ} \cos \theta$ .

\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)} = \frac{1 - \tan 45^{\circ} \tan \theta}{1 + \tan 45^{\circ} \tan \theta}

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it using the given value of $\theta = 30^{\circ}$ .

\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)} = \frac{1 - \tan 45^{\circ} \tan 30^{\circ}}{1 + \tan 45^{\circ} \tan 30^{\circ}}

We know that $\tan 45^{\circ} = 1$ and $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$ . Substituting these values into the expression, we get:

\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}}

Simplifying this expression further, we get:

\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}

In this article, we evaluated the trigonometric expression $\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)}$ without using tables. We used trigonometric identities and formulas to simplify the expression and find its value. The final value of the expression is $\frac{\sqrt{3} - 1}{\sqrt{3} + 1}$ .
Q&A: Evaluating Trigonometric Expressions without Using Tables

In our previous article, we evaluated the trigonometric expression $\frac{\cos (45^{\circ} + \theta)}{\cos (45^{\circ} - \theta)}$ without using tables. We used trigonometric identities and formulas to simplify the expression and find its value. In this article, we will answer some frequently asked questions related to evaluating trigonometric expressions without using tables.

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

A: There are several common trigonometric identities that can be used to simplify expressions. Some of these identities include:

Angle Addition Formula: $\cos (A + B) = \cos A \cos B - \sin A \sin B$
Angle Subtraction Formula: $\cos (A - B) = \cos A \cos B + \sin A \sin B$
Pythagorean Identity: $\sin^2 A + \cos^2 A = 1$
Double Angle Formula: $\cos 2A = 2\cos^2 A - 1$

These identities can be used to simplify a wide range of trigonometric expressions.

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 need to identify the two angles that are being added together. Then, you can use the formula to expand the expression.

For example, if you have the expression $\cos (A + B)$ , you can use the angle addition formula to expand it as follows:

\cos (A + B) = \cos A \cos B - \sin A \sin B

Q: What is the difference between the angle addition and angle subtraction formulas?

A: The angle addition formula is used to simplify expressions where two angles are being added together, while the angle subtraction formula is used to simplify expressions where two angles are being subtracted.

For example, if you have the expression $\cos (A - B)$ , you can use the angle subtraction formula to expand it as follows:

\cos (A - B) = \cos A \cos B + \sin A \sin B

Q: How can I use the Pythagorean identity to simplify an expression?

A: To use the Pythagorean identity to simplify an expression, you need to identify the trigonometric functions that are being squared. Then, you can use the identity to simplify the expression.

For example, if you have the expression $\sin^2 A + \cos^2 A$ , you can use the Pythagorean identity to simplify it as follows:

\sin^2 A + \cos^2 A = 1

Q: What is the significance of the double angle formula?

A: The double angle formula is used to simplify expressions where the angle is being doubled. It is a useful formula for simplifying expressions that involve the cosine function.

For example, if you have the expression $\cos 2A$ , you can use the double angle formula to simplify it as follows:

\cos 2A = 2\cos^2 A - 1

In this article, we answered some frequently asked questions related to evaluating trigonometric expressions without using tables. We discussed some common trigonometric identities and formulas that can be used to simplify expressions, and provided examples of how to use these identities to simplify expressions. We hope that this article has been helpful in providing a better understanding of trigonometric expressions and how to simplify them without using tables.