# Thread: [SOLVED] Cosine of the Sum of Two Angles

1. ## [SOLVED] Cosine of the Sum of Two Angles

Given sin $\theta$ = $\frac{-5}{13}$ in quadrant III, tan $\phi$ = $\frac{-8}{15}$ in quadrant II; find the value of sin( $\theta$ + $\phi$):

I've posted my solution -- which the answer key says is wrong -- here, http://imagebin.ca/img/tnINM0J.png .

Please tell me what I'm doing wrong.

2. Originally Posted by beardedoneder
Given sin $\theta$ = $\frac{-5}{13}$ in quadrant III, tan $\phi$ = $\frac{-8}{15}$ in quadrant II; find the value of sin( $\theta$ + $\phi$):

I've posted my solution -- which the answer key says is wrong -- here, http://imagebin.ca/img/tnINM0J.png .

Please tell me what I'm doing wrong.
first if tan $\phi$ = $\frac{-8}{15}$is in Q2
or actually should be $\tan\phi= \frac{8}{-15}$ if it is in Q2
then $\sin\phi = \frac{8}{17}$

therefore

$\sin(\theta + \phi)=\left(\text{-}\frac{5}{13}\right)\left(\text{-}\frac{15}{17}\right) + \left(\text{-}\frac{12}{13}\right)\left(\frac{8}{15}\right)
\Rightarrow
\frac{75}{221} - \frac{96}{221}
\Rightarrow
-\frac{21}{221}$

just got too late with latex

3. Originally Posted by beardedoneder
Given sin $\theta$ = $\frac{-5}{13}$ in quadrant III, tan $\phi$ = $\frac{-8}{15}$ in quadrant II; find the value of sin( $\theta$ + $\phi$):

I've posted my solution -- which the answer key says is wrong -- here, http://imagebin.ca/img/tnINM0J.png .

Please tell me what I'm doing wrong.

4. I would do it like this

Originally Posted by beardedoneder
Given sin $\theta$ = $\frac{-5}{13}$ in quadrant III,
use $\sin^2\theta+\cos^2\theta = 1$ to find $\cos\theta$

Originally Posted by beardedoneder
tan $\phi$ = $\frac{-8}{15}$ in quadrant II;
use $1+\tan^2\phi = \sec^2\phi$ to find $\cos\phi$

and then $\sin^2\phi+\cos^2\phi = 1$ to find $\sin\phi$

5. Hello, beardedoneder!

Given: . $\begin{array}{ccc}\sin\theta \:=\:\text{-}\frac{5}{13}& \text{in quadrant III} \\ \\ [-3mm]
\tan\phi \:=\:\text{-}\frac{8}{15} & \text{in quadrant II}\end{array}$

find the value of: . $\sin(\theta + \phi)$

We have: . $\sin\theta \:=\:-\frac{5}{13} \:=\:\frac{opp}{hyp}$

$\theta$ is in a right triangle with: $opp = -5,\;hyp = 13$

Pythagorus says: . $adj = \pm12$

. . In Quadrant III: $adj = -12$

Hence: . $\begin{Bmatrix} \sin\theta &=& \text{-}\dfrac{5}{13} \\ \\[-3mm] \cos\theta &=& \text{-}\dfrac{12}{13} \end{Bmatrix}$ .[1]

We have: . $\tan \phi \:=\:-\frac{8}{15} \:=\:\frac{opp}{adj}$

$\phi$ is in Quadrant II with: $opp = 8,\;adj = -15$

Pythagorus says: . $hyp = 17$

Hence: . $\begin{Bmatrix}\sin\phi &=& \dfrac{8}{17} \\ \\[-3mm] \cos\phi &=& \text{-}\dfrac{15}{17} \end{Bmatrix}$ .[2]

We want: . $\sin(\theta + \phi) \;=\;\sin\theta\cos\phi + \cos\theta\sin\phi$

Substitute [1] and [2]:

$\sin(\theta + \phi) \;=\;\left(\text{-}\frac{5}{13}\right)\left(\text{-}\frac{15}{17}\right) + \left(\text{-}\frac{12}{13}\right)\left(\frac{8}{15}\right) \;=\;\frac{75}{221} - \frac{96}{221} \;=\;-\frac{21}{221}$

6. ## Answer Key Was Wrong

Originally Posted by beardedoneder
Given sin $\theta$ = $\frac{-5}{13}$ in quadrant III, tan $\phi$ = $\frac{-8}{15}$ in quadrant II; find the value of sin( $\theta$ + $\phi$):

I've posted my solution -- which the answer key says is wrong -- here, http://imagebin.ca/img/tnINM0J.png .

Please tell me what I'm doing wrong.
I got a copy of the Solution Key. It solves the problem the same way I do, but it gives a product of $\frac{225}{221}$ for $-\frac{5}{13} \cdot -\frac{15}{17}$ instead of $\frac{75}{221}$. Which is why the answer key gives an answer of $\frac{129}{221}$, instead of $-\frac{21}{221}$. My solution and answer were correct.

7. ## Did you mean to say, ...

Originally Posted by bigwave
first if tan $\phi$ = $\frac{-8}{15}$is in Q2
or actually should be $\tan\phi= \frac{8}{-15}$ if it is in Q2
then $\sin\phi = \frac{8}{17}$

therefore

$\sin(\theta + \phi)=\left(\text{-}\frac{5}{13}\right)\left(\text{-}\frac{15}{17}\right) + \left(\text{-}\frac{12}{13}\right)\left(\frac{8}{15}\right)
\Rightarrow
\frac{75}{221} - \frac{96}{221}
\Rightarrow
-\frac{21}{221}$

just got too late with latex

Did you mean to say, $-\frac{12}{13} \cdot \frac{8}{17}$ ?

8. ## Did you mean to say, ...

Originally Posted by Soroban
Hello, beardedoneder!

Substitute [1] and [2]:

$\sin(\theta + \phi) \;=\;\left(\text{-}\frac{5}{13}\right)\left(\text{-}\frac{15}{17}\right) + \left(\text{-}\frac{12}{13}\right)\left(\frac{8}{15}\right) \;=\;\frac{75}{221} - \frac{96}{221} \;=\;-\frac{21}{221}$

Did you mean to say, $-\frac{12}{13} \cdot \frac{8}{17}$ ?