Question

You are given that cos(A)=15/17, with A in Quadrant IV, and cos(B)=4/5, with B in Quadrant IV. Find cos(A−B). Give your answer as a fraction.

Please help i need this asap.

Answer 1

Answer:

4/5

Step-by-step explanation:

Answer 2

There are four quadrants in a coordinate geometry.

The value of cos(A - B) is 84/85

The given parameters are:

$\mathbf{cos(A) = \frac{15}{17}}$

$\mathbf{cos(B) = \frac{4}{5}}$

Using trigonometry ratio, we have:

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

Substitute $\mathbf{cos(A) = \frac{15}{17}}$

$\mathbf{sin^2(A) + (\frac {15}{17})^2 =1}$

Expand

$\mathbf{sin^2(A) + \frac{225}{289} =1}$

Collect like terms

$\mathbf{sin^2(A) =1 - \frac{225}{289}}$

Evaluate fraction

$\mathbf{sin^2(A) =\frac{64}{289}}$

Take square roots of both sides

$\mathbf{sin(A) =\pm\frac{8}{17}}$

The angle is in the fourth quadrant.

So, we have:

$\mathbf{sin(A) =-\frac{8}{17}}$

Also,

Substitute $\mathbf{cos(B) = \frac{4}{5}}$ in $\mathbf{sin^2(B) + cos^2(B)= 1}$

$\mathbf{sin^2(B) + (\frac 45)^2= 1}$

$\mathbf{sin^2(B) + \frac{16}{25}= 1}$

Collect like terms

$\mathbf{sin^2(B) = 1 - \frac{16}{25}}$

Simplify fraction

$\mathbf{sin^2(B) = \frac{9}{25}}$

Take square roots

$\mathbf{sin(B) = \pm \frac{3}{5}}$

The angle is also in fourth quadrant

So, we have:

$\mathbf{sin(B) = - \frac{3}{5}}$

To calculate cos(A - B), we make use of:

$\mathbf{cos(A - B) = cos(A)cos(B) + sin(A)sin(B)}$

So, we have:

$\mathbf{cos(A - B) = \frac{15}{17} \times \frac{4}{5} + \frac{-8}{17} \times \frac{-3}{5}}$

Evaluate

$\mathbf{cos(A - B) = \frac{60}{85} + \frac{24}{85}}$

Take LCM

$\mathbf{cos(A - B) = \frac{60+24}{85}}$

$\mathbf{cos(A - B) = \frac{84}{85}}$

Hence, the value of cos(A - B) is 84/85

Read more about quadrants at:

https://brainly.com/question/7196054