Calculating Trigonometric Expressions Using Tangent Values

In this article, we will explore how to calculate a trigonometric expression using the values of tangent for two angles. We will delve into the steps and derive the exact value of the cosine and sine of these angles to simplify the expression cos B - sin B.

Given Information

Given the values of tangent for two angles, tan A 1 and tan B √3. Our goal is to find the value of the expression cos B - sin B.

Step 1: Determine the Angles A and B

Since tan A 1, we can deduce that the angle A is 45°. The tangent of 45° is 1. Similarly, since tan B √3, angle B is 60°, as the tangent of 60° is √3.

Step 2: Find the Cosine and Sine Values

Knowing that A 45° and B 60°, we can now find the values of cosine and sine for these angles:

For A 45°: cos 45° sin 45° 1/√2 For B 60°: cos 60° 1/2 and sin 60° √3/2

Step 3: Calculate the Expression

Now, we substitute the values of cosine and sine into the expression cos B - sin B:

begin{align*} text{cos B - sin B} cos 45°. cos 60° - sin 45°. sin 60° left(frac{1}{sqrt{2}}right) left(frac{1}{2}right) - left(frac{1}{sqrt{2}}right) left(frac{sqrt{3}}{2}right) frac{1}{2sqrt{2}} - frac{sqrt{3}}{2sqrt{2}} frac{1 - sqrt{3}}{2sqrt{2}} end{align*}

Conclusion

The given expression cos B - sin B, when A 45° and B 60°, is equal to 1 - √3 divided by 2√2. This can be simplified to 1 - √3 over 2√2, or 1/2 - √3/2√2.

Calculator Integration for Trigonometric Functions

For those who prefer a more straightforward method, using a trigonometric calculator can be a great aid. By entering the tangent values and utilizing the inverse tangent function, you can directly obtain the corresponding angles and then calculate the desired trigonometric expression.

Tips for Solving Trigonometric Expressions

Remember the common tangent values for standard angles like 0°, 30°, 45°, 60°, and 90°. Use the Pythagorean identities to relate sine and cosine. Consider using a trigonometric table or a calculator for precision.

Further Reading

If you are interested in learning more about trigonometric functions and expressions, you can explore resources on websites like Khan Academy, Mathway, or Wolfram Alpha. These platforms offer extensive tutorials and practice problems to improve your understanding of trigonometry.