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Chapter 5: Problem 52

In Exercises \(51-56,\) find the exact value of the trigonometric function given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III. \(\tan (u-v)\)

Short Answer

Expert verified
The exact value of \( \sin\left( u + v\right) \) is \( \frac{4}{5} \).

Step by step solution

01

Find Cosine and Sine of Given Angles

Firstly, let鈥檚 find the unknown trigonometric functions \( \cos u \) and \( \sin v \). As both the angles are in Quadrant III, both cosine and sine are negative. We can apply the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find these. So, for angle \( u \), \( \cos u = - \sqrt{1- (\sin u)^2} = -\sqrt{1 - \left(-\frac{7}{25}\right)^2} = -\frac{24}{25} \) and for angle \( v \), \( \sin v = - \sqrt{1 - (\cos v)^2} = -\sqrt{1 - \left(-\frac{4}{5}\right)^2} = -\frac{3}{5} \).
02

Apply Sine of a Sum Formula

The formula for the sine of a sum of two angles is \( \sin\left(a + b\right) = \sin(a)\cos(b) + \cos(a)\sin(b) \). Apply the values calculated in the previous step into this formula for \( \sin\left( u + v\right) \). Substituting the given and calculated values, we get: \( \sin\left( u + v\right) = \sin(u)\cos(v) + \cos(u)\sin(v) = \left(-\frac{7}{25}\right)\left(-\frac{4}{5}\right) + \left(-\frac{24}{25}\right)\left(-\frac{3}{5}\right) \).
03

Simplify The Expression

To simplify the above expression, multiply the fractions together: \( \sin\left( u + v\right) = \frac{28}{125} + \frac{72}{125} \). Now, add these fractions: \( \sin\left( u + v\right) = \frac{100}{125} \) which simplifies to \( \frac{4}{5} \). So, the exact value of \( \sin\left(u + v\right) \) is \( \frac{4}{5} \).

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Most popular questions from this chapter

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( \cos 2x + \sin x = 0 \)

In Exercises 111 - 124, verify the identity. \( \sin \dfrac{\alpha}{3} \cos \dfrac{\alpha}{3} = \dfrac{1}{2} \sin \dfrac{2\alpha}{3} \)

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{\pi}{8} \)

In Exercises 67-72, (a) determine the quadrant in which \( u/2 \) lies, and (b) find the exact values of \( \sin(u/2) \), \( \cos(u/2) \), and \( \tan(u/2) \) using the half-angle formulas. \( \sec u = \dfrac{7}{2}, \dfrac{3\pi}{2} < u < 2\pi \)

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \tan^2 2x \cos^4 2x \)
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