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Chapter 5: Problem 47

In Exercises \(43-50\) , find the exact value of the trigonometric function given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) Both \(u\) and \(v\) are in Quadrant II.) \(\tan (u+v)\)

Short Answer

Expert verified
The exact value of \( \tan(u+v) \) is \( \frac{33}{56} \).

Step by step solution

01

Identify the sine and cosine of the given angles

The sine of u, that is \( \sin u \), is given as \( \frac{5}{13} \). Using the Pythagorean identity \( \sin^2u + \cos^2u = 1 \), we can solve for \( \cos u \) which gives us \( \cos u = \sqrt{1 - \sin^2 u} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = - \frac{12}{13} \). The negative value is chosen because cosine is negative in quadrant II. Similarly, given the cosine of v, that is \( \cos v = - \frac{3}{5} \), we find \( \sin v \) using the Pythagorean identity to get \( \sin v = - \sqrt{1 - \cos^2 v} = - \sqrt{1 - \left(- \frac{3}{5}\right)^2} = - \frac{4}{5} \). The negative value is chosen because sine is negative in quadrant II.
02

Apply the formula for the tangent of the sum of two angles

The formula for the tangent of the sum of two angles states that \( \tan(u+v) = \frac{\sin u \cos v + \cos u \sin v}{\cos u \cos v - \sin u \sin v} \). In this case we substitute the given values of \( \sin u, \sin v, \cos u, \cos v \) into the formula which gives us: \( \tan(u+v) = \frac{\frac{5}{13} \times - \frac{3}{5} + - \frac{12}{13} \times -\frac{4}{5}}{- \frac{12}{13} \times - \frac{3}{5} - \frac{5}{13} \times -\frac{4}{5}} \).
03

Simplify your equation

Solve the equation to get the final answer. After simplifying, \( \tan(u+v) = \frac{33}{56} \).

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Most popular questions from this chapter

In Exercises 139 and 140, determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \( u, \sin 2u = -2 \sin u \cos u \).

In Exercises 111 - 124, verify the identity. \( \cos \left(\dfrac{\pi}{3} + x\right) x \cos\left(\dfrac{\pi}{3} - x\right) = \cos x \)

In Exercises 99 - 102, use the sum-to-product formulas to find the exact value of the expression. \( \cos 120^\circ + \cos 60^\circ \)

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