If tan𝜽= 𝟑/𝟒, find the values of secθ and cosθ.

Solution

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

Given tan(θ) = 3/4, we can label the opposite side of the right triangle as 3 and the adjacent side as 4.

Using the Pythagorean theorem, we can find the hypotenuse:

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

25 = c²

c = 5

Now we can use the definitions of the trigonometric functions to find the values of sec(θ) and cos(θ):

sec(θ) = hypotenuse/adjacent = 5/4

cos(θ) = adjacent/hypotenuse = 4/5

Therefore, sec(θ) = 5/4 and cos(θ) = 4/5.