Question

Given that sin alpha=3/5,cos beta=4/5 , alpha and beta are in 1st quadrant then sin ( alpha+beta)=

Answer 1

The value of sin (alpha + beta) is $\frac{24}{25}$.

Explanation:

According to the given information, the value of sin alpha is given as 3/5 and the value of cos beta is given as 4/5, where the angles alpha and beta are present in the first quadrant. We need to find the value of (alpha+beta).

Now, we know that, the well known formula of the addition of two angles in the first quadrant under sine, that is,

sin (alpha + beta) = sin (alpha) cos (beta) + cos (alpha) sin (beta)...(1)

Now, given that, the value of sin alpha is given as 3/5.

Now, we know that, the well-known trigonometrical identity, that is, sin²(alpha) + cos²(alpha) = 1.

Or, cos²(alpha) = 1 - sin²(alpha)

Or, cos(alpha) = √(1 - sin²(alpha))

Then,, putting the value of sin alpha, we get,

cos (alpha) = √(1 - (3/5)²)

Or, cos (alpha) = √(1 - 9/25)

Or, cos (alpha) = √($\frac{25-9}{25}$)

Or, cos (alpha) = √(16/25) = 4/5

Proceeding in a similar manner, we get,

sin (beta) = √(1-(4/5)²)

Or, sin (beta) = √(1-$\frac{16}{25}$)

Or, sin (beta) = √($\frac{9}{25}$)

Or, sin (beta) =3/5

Then, (1) becomes,

sin (alpha + beta) = sin (alpha) cos (beta) + cos (alpha) sin (beta)

=$\frac{3}{5}\frac{4}{5} + \frac{4}{5}\frac{3}{5} \\=\frac{12}{25}+ \frac{12}{25}\\=\frac{24}{25}$

Thus, the value of sin (alpha + beta) is $\frac{24}{25}$.

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Answer 2

Answer:

Explanation:

The value of sin (alpha + beta) is .

Explanation:

According to the given information, the value of sin alpha is given as 3/5 and the value of cos beta is given as 4/5, where the angles alpha and beta are present in the first quadrant. We need to find the value of (alpha+beta).

Now, we know that, the well known formula of the addition of two angles in the first quadrant under sine, that is,

sin (alpha + beta) = sin (alpha) cos (beta) + cos (alpha) sin (beta)...(1)

Now, given that, the value of sin alpha is given as 3/5.

Now, we know that, the well-known trigonometrical identity, that is, sin²(alpha) + cos²(alpha) = 1.

Or, cos²(alpha) = 1 - sin²(alpha)

Or, cos(alpha) = √(1 - sin²(alpha))

Then,, putting the value of sin alpha, we get,

cos (alpha) = √(1 - (3/5)²)

Or, cos (alpha) = √(1 - 9/25)

Or, cos (alpha) = √()

Or, cos (alpha) = √(16/25) = 4/5

Proceeding in a similar manner, we get,

sin (beta) = √(1-(4/5)²)

Or, sin (beta) = √(1-)

Or, sin (beta) = √()

Or, sin (beta) =3/5

Then, (1) becomes,

sin (alpha + beta) = sin (alpha) cos (beta) + cos (alpha) sin (beta)

=

Thus, the value of sin (alpha + beta) is 24/25