If Sin𝜽= 𝟕/𝟐𝟓, find the values of cosθ and tanθ.

Solution

We know that sin(θ) = opposite/hypotenuse in a right triangle, where θ is one of the acute angles of the triangle.

Given sin(θ) = 7/25, we can label the opposite side of the right triangle as 7 and the hypotenuse as 25.

Using the Pythagorean theorem, we can find the adjacent side:

a² + b² = c²

a² + 7² = 25²

a² + 49 = 625

a² = 576

a = ±24

Since cos(θ) = adjacent/hypotenuse in a right triangle, and we know that θ is in the first or second quadrant (where cos(θ) is positive), we can use the positive value of adjacent side to find the value of cos(θ):

cos(θ) = 24/25

Finally, we can use the definitions of the trigonometric functions to find the value of tan(θ):

tan(θ) = sin(θ)/cos(θ) = (7/25)/(24/25) = 7/24

Therefore, cos(θ) = 24/25 and tan(θ) = 7/24.