Question

Find the exact value of each of the following under the given conditions.
a. cosine left parenthesis alpha plus beta right parenthesis        b. sine left parenthesis alpha plus beta right parenthesis        c. tangent left parenthesis alpha plus beta right parenthesis
tangent alpha equals one half
​, pi less than alpha less than StartFraction 3 pi Over 2 EndFraction
​, and cosine beta equals three fifths
​, StartFraction 3 pi Over 2 EndFraction less than beta less than 2 pi

Answer 1

Answer:

$(a)-\dfrac{11\sqrt{5}}{25} \\(b) -\dfrac{2\sqrt{5}}{25} \\(c)\dfrac{11}{2}$

Step-by-step explanation:

$\tan \alpha =\dfrac12, \pi < \alpha< \dfrac{3 \pi}{2}$

Therefore:

$\alpha is in Quadrant III$

Opposite = -1

Adjacent =-2

Using Pythagoras Theorem

$Hypotenuse^2=Opposite^2+Adjacent^2\\=(-1)^2+(-2)^2=5\\Hypotenuse=\sqrt{5}$

Therefore:

$\sin \alpha =-\dfrac{1}{\sqrt{5}}\\\cos \alpha =-\dfrac{2}{\sqrt{5}}$

Similarly

$\cos \beta =\dfrac35, \dfrac{3 \pi}{2}<\beta<2\pi\\\beta is in Quadrant IV (x is negative, y is positive), therefore:\\Adjacent= -3\\ Hypotenuse=5\\Opposite=4 (Using Pythagoras Theorem)$

$\sin \beta =\dfrac{4}{5}\\\tan \beta =-\dfrac{4}{3}$

(a)

$\cos(\alpha + \beta)=\cos\alpha\cos\beta-\sin \alpha\sin \beta$

$=-\dfrac{2}{\sqrt{5}}\cdot \dfrac{3}{5}-(-\dfrac{1}{\sqrt{5}})(\dfrac{4}{5})\\=-\dfrac{2\sqrt{5}}{5}\cdot \dfrac{3}{5}+\dfrac{\sqrt{5}}{5}\cdot\dfrac{4}{5}\\=-\dfrac{2\sqrt{5}}{25}$

(b)

$\sin(\alpha + \beta)=\sin\alpha\cos\beta+\cos \alpha\sin \beta$

$\sin(\alpha + \beta)=\sin\alpha\cos\beta+\cos \alpha\sin \beta\\=-\dfrac{1}{\sqrt{5}}\cdot\dfrac35+(-\dfrac{2}{\sqrt{5}})(\dfrac{4}{5})\\=-\dfrac{\sqrt{5}}{5}\cdot\dfrac35-\dfrac{2\sqrt{5}}{5}\cdot\dfrac{4}{5}\\=-\dfrac{11\sqrt{5}}{25}$

(c)

$\tan(\alpha + \beta)=\dfrac{\sin(\alpha + \beta)}{\sin(\alpha + \beta)}=-\dfrac{11\sqrt{5}}{25} \div -\dfrac{2\sqrt{5}}{25} =\dfrac{11}{2}$