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In triangle ABC, right-angled at B, if $t a n A = 13$ , then what is the value of sin A cos C +cos A sin C?

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Detailed Solution

Given that, triangle ABC is right-angled at B and

t a n A = 13

.
By the definition of tangent of an angle, we have

tan A = 13 = P B

Let P = k and B = k

3

for some constant k.
Using Pythagoras theorem, we have

H 2 = P 2 + B 2 H = 3 k 2 + k 2 H = 4 k 2 H = 2 k

Now,
The triangle is right angled at B.
We know that the sum of the interior angles of a triangle is 180.

⇒

A+B+C=180°

⇒

A + C=180°-90° [B = 90°]

⇒

A + C = 90°

⇒

C = 90° - A………[1]
We need to find the value of the expression,
sin A cos C +cos A sin C.

⇒

sin A cos(90°-A) + cos A sin(90°-A) (since, eq(1))
We know that cos(90° – A) = sin A and sin(90° – A) = cos A.

⇒

sin A sin A + cos A cos A

⇒ sin 2 A + cos 2 A N o w, sin A = P H ⇒ sin A = k 2 k = 12 cos A = B H ⇒ cos A = k 3 2 k = 32

Substituting the values in the expression, we get

= 122 + 322 ⇒ 14 + 34 = 1

Therefore, the value of the expression sin A cos C + cos A sin C is 1.

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