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Given cos a= 3/5, sin B = 5/13 and a and B are first quadrant angles find cos(a + B)

A. 1/3
B. 16/65
C. -33/65

Please explain

User Drdrdr
by
8.4k points

1 Answer

3 votes

Answer:

Option B is correct.

Value of cos(a+B) is,
(16)/(65)

Explanation:

Using the formula:


\cos(a+B) = \cos a \cos B -\sin a \sin B

Given the values:


\cos a = (3)/(5) and
\sin B = (5)/(13)

Using trigonometric identity:


\sin^2 \theta + \cos^2 \theta =1

or


\sin \theta = √(1-\cos^2 \theta)

or


\cos \theta = √(1-\sin^2 \theta)

Find the value of sin a and cos B;

Using trigonometric identity:


\sin a = √(1-\cos^2 a)


\sin a = \sqrt{1-((3)/(5))^2} =
\sqrt{1-(9)/(25)} = \sqrt{(25-9)/(25)}=\sqrt{(16)/(25) } =(4)/(5)

Similarly;


\cos B = √(1-\sin^2 B)


\cos B = \sqrt{1-((5)/(13))^2} =
\sqrt{1-(25)/(169)} = \sqrt{(169-25)/(169)}=\sqrt{(144)/(169) } =(12)/(13)

Substitute these given values equation [1] we have;


\cos(a+B) =(3)/(5) \cdot (12)/(13)-(4)/(5) \cdot (5)/(13)


\cos(a+B) =(36)/(65)-(20)/(65)

then;


\cos(a+B) =(36-20)/(65)=(16)/(65)

Since, the angles a and B are in first quadrant angles therefore the value of cos(a+B) is,
(16)/(65)


User Nikolay Lukyanchuk
by
7.6k points

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