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Chapter 6: Problem 85

Use words to describe the formula for each of the following: the tangent of the difference of two angles.

Short Answer

Expert verified
The tangent of the difference of two angles A and B is given by the formula \(\tan(A-B) = \frac{\tan(A) - \tan(B)}{1+\tan(A)\tan(B)}\). It represents the tangent of the angular separation between A and B.

Step by step solution

01

Explaining the concept of Tangent

The tangent of an angle in a right triangle is the ratio of the side opposite to the angle to the side adjacent to the angle. In terms of the unit circle, it is the y-coordinate divided by the x-coordinate of a point on the circle.
02

Explaining the concept of Difference of Two Angles

The difference between two angles A and B is obtained by subtracting B from A. It represents the angular separation between A and B, where A > B.
03

Describing the formula for the Tangent of the Difference of Two Angles

The formula for the tangent of the difference of two angles A and B is given by \(\tan(A-B) = \frac{\tan(A) - \tan(B)}{1+\tan(A)\tan(B)}\). This formula gives the tangent of the angle that is obtained by subtracting B from A.

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Most popular questions from this chapter

Suppose you are solving equations in the interval \([0,2 \pi)\) Without actually solving equations, what is the difference between the number of solutions of \(\sin x=\frac{1}{2}\) and \(\sin 2 x=\frac{1}{2} ?\) How do you account for this difference?

Exercises \(110-112\) will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 110 to answer each of the following. a. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} ?\) b. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} \cos 30^{\circ} ?\)

Use words to describe the formula for each of the following: the sine of the difference of two angles.

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$ \cos ^{2} x-\cos x-1=0 $$

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$ \sin 2 x=2-x^{2} $$
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