Question

Given cos ⁡ ( θ ) = 5/6 , then sin ⁡ ( θ ) = a/b .

Determine the values of a and b.

Answer 1

Answer:

a = √11 and b = 6

Step-by-step explanation:

Refer to attached picture for reference

for an right triangle with angle θ

we are given

cos θ = 5/6 = length of adjacent side / length of hypotenuse

hence

adjacent length = 5 units

hypotenuse length = 6 units

the missing side is the "opposite" length which we can find with the Pythagorean equation. in our case:

hypotenuse ² = adjacent ² + opposite²   (rearrange)

opposite ² = hypotenuse ² - adjacent ²

opposite ² = 6² - 5²

opposite = √ (6²-5²) = √11

sin θ = opposite length / hypotenuse  (substitute values above)

sin θ = √11 / 6

hence a = √11 and b = 6

Answer 2

The values of a and b are $\mathbf{a = \sqrt{11}}$ and $\mathbf{b = 6}}$

The given parameters are:

$\mathbf{cos(\theta) = \frac 56}$

$\mathbf{sin(\theta) = \frac ab}$

To determine the values of a and b, we make use of the following trigonometry ratio

$\mathbf{cos^2(\theta) + sin^2(\theta) = 1}$

Substitute values for cosine and sine

$\mathbf{(\frac 56)^2 + (\frac ab)^2 = 1}$

Evaluate the exponents

$\mathbf{\frac{25}{36} + (\frac ab)^2 = 1}$

Collect like terms

$\mathbf{(\frac ab)^2 = 1 - \frac{25}{36}}$

Take LCM

$\mathbf{(\frac ab)^2 = \frac{36 -25}{36}}$

$\mathbf{(\frac ab)^2 = \frac{11}{36}}$

Take square roots of both sides

$\mathbf{\frac ab = \frac{\sqrt{11}}{6}}$

By comparison:

$\mathbf{a = \sqrt{11}}$

$\mathbf{b = 6}}$

Read more about trigonometry ratios at:

https://brainly.com/question/24888715