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snow_tiger [21]
3 years ago
10

For positive acute angles A and B, it is known that cos A= 60/61 and tan B= 45/28. Find the value of sin(A+B) in simplest form

Mathematics
1 answer:
inna [77]3 years ago
7 0

Answer:

\displaystyle \sin(A+B)=\frac{3008}{3233}

Step-by-step explanation:

We are given that:

\displaystyle \cos(A)=\frac{60}{61}\text{ and } \tan(B)=\frac{45}{28}

And that the two angles <em>A</em> and <em>B</em> are both in QI (positive acute angles).

Given this, we want to determine the value of sin(A + B).

Recall that cosine is the ratio of the adjacent side to the hypotenuse. Using this information, we can determine the opposite side with respect to <em>A:</em>

<em />\displaystyle o=\sqrt{61^2-60^2}=\sqrt{121}=11<em />

<em />

Likewise, for <em>B</em>, tangent gives the ratio of the opposite side to the adjacent. Using this information, we can determine the hypotenuse of <em>B:</em>

<em />h=\sqrt{45^2+28^2}=\sqrt{2809}=53<em />

<em />

In conclusion:

With respect to <em>A</em>, the adjacent side is 60, the opposite side is 11, and the hypotenuse is 61.

With respect to <em>B, </em>the adjacent side is 28, the opposite side is 45, and the hypotenuse is 53.

We can rewrite our original expression as:

\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)

And since <em>A</em> and <em>B</em> are both in QI, all trigonometric ratios will be positive.

Using the above conclusions, substitute:

\displaystyle \sin(A+B)=\left(\frac{11}{61}\right)\left(\frac{28}{53}\right)+\left(\frac{45}{53}\right)\left(\frac{60}{61}\right)

Simplify. Hence:

\displaystyle \sin(A+B)=\frac{308+2700}{3233} = \frac{3008}{3233}

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