Question

the value of tan(alpha+beta) given cos(alpha+beta)= -528/697 and sin(alpha+beta)= 455/697 and sin(a)=40/41 and sin(b)=15/17

Answer 1

PROBLEM STATEMENT

To evaluate the value of

`\tan (\alpha+\beta)`

GIVEN

`\begin{gathered} \cos (\alpha+\beta)=-(528)/(697) \\ \sin (\alpha+\beta)=(455)/(697) \\ \sin (\alpha)=(40)/(41) \\ \sin (\beta)=(15)/(17) \end{gathered}`

SOLUTION

Recall the trigonometric identity:

`\tan (x)=(\sin (x))/(\cos (x))`

If we have

`x=\alpha+\beta`

Therefore, we have that:

`\tan (\alpha+\beta)=(\sin(\alpha+\beta))/(\cos(\alpha+\beta))`

Substituting for the values of sin and cos, we have:

`\tan (\alpha+\beta)=((455)/(697))/(-(528)/(697))`

Rewriting, we have:

`\begin{gathered} \tan (\alpha+\beta)=-(455)/(697)/(528)/(697) \\ \tan (\alpha+\beta)=-(455)/(697)*(697)/(528) \\ \tan (\alpha+\beta)=-(455)/(528) \end{gathered}`

ANSWER

`\tan (\alpha+\beta)=-(455)/(528)`