Mathematics, 02.04.2020 03:37 dmperez1213

If sinA = 5/12 and 90° < A < 180°, and tanB = 5 and 180° < B < 270°, determine the exact value of cos(B – A).

Answers: 2

Show answers

Answers

Answer from: bl6659

Step-by-step explanation:

The trig identity for cos(B - A) is as follows:

cos(B)cos(A) + sin(B)sin(A)

We were given the sin(A) and told it was in QII (that's the quadrant where an angle is between 90 and 180), and we were given tan(B) and told it was in QIII (that's the quadrant where both sin and cos are negative. Remember that the tangent ratio is the same thing as sin/cos).

We can use the sin(A) to find the cos(A) by setting up a reference angle in standard position (at the origin) and using Pythagorean's Theorem to find the missing side. The sin ratio by definition is sin(A) = side opposite A/hypotenuse, and the cos ratio is cos(A) = side adjacent A/hypotenuse.

Using the fact that the sin(A) = 5/12, that tells us that the hypotenuse is 12, the height of the triangle is 5, and we need to find the side adjacent to A. The cos of A will be negative since cos is negative in QII.

12^2=5^2+x^2 and

144=25+x^2 and

119=x^2 so

$x=\sqrt{119}$

That means that the cos(A) is

$cos(A)=-\frac{\sqrt{119} }{12}$

Now for B. We will set the reference angle in standard position (at the origin) in QIII. If the tan(B) = 5 and the tan ratio by definition is side opposite B/side adjacent to B, then in order for the tan ratio to be ust a 5, that means that it was originally 5/1, or to be more specific, -5/-1, with -5 being the height of the right triangle and -1 being the side adjacent to the reference angle. We need to find the length of the hypotenuse in order to find either the sin(B) or the cos(B) since both of those ratios need the hypotenuse. We will again use Pythagorean's Theorem to find the hypotenuse:

c^2=(-5)^2+(-1)^2 and

c^2=25+1 and

$c=\sqrt{26}$

That puts the

$sin(B)=-\frac{5}{\sqrt{26} }$ but we'll rationalize that denominator:

$sin(B)=-\frac{5}{\sqrt{26} }*\frac{\sqrt{26} }{\sqrt{26} }$ which simplifies to

$sin(B)=-\frac{5\sqrt{26} }{26}$

That puts the

$cos(B)=-\frac{1}{\sqrt{26} }*\frac{\sqrt{26} }{\sqrt{26} }$ which simplifies to

$cos(B)=-\frac{\sqrt{26} }{26}$

Our values are, altogether then:

$sin(A)=\frac{5}{12}$ , $cos(A)=-\frac{\sqrt{119} }{12}$ , $sin(B)=-\frac{5\sqrt{26} }{26}$ , and $cos(B)=-\frac{\sqrt{26} }{26}$

Now we will put all those values where they belong in the identity cos(B - A):

$cos(B-A)=(-\frac{\sqrt{26} }{26}*(-\frac{\sqrt{119} }{12})+(-\frac{5\sqrt{26} }{26}*\frac{5}{12})$ Now we need to simplify that:

$cos(B-A)=\frac{\sqrt{3094} - 25\sqrt{26} }{312}$

And, believe it or not, that's as simple as it gets. Yikes! That's the exact value. If you want the decimal value, which is not considered exact, you'd throw that into your calculator and get an irrational number.

Answer from: Quest

Most of the people don’t make the sense of taking it

Answer from: Quest

With out knowing the functions you have to chose form, this could be one.y = 5.00xx = hours worked and y = the money he will earn.

Another question on Mathematics

Mathematics, 21.06.2019 14:50

For f(x)=2x+1 and g(x)= x^2 -7, find (f•g)(x)

Answers: 1

Answer

Mathematics, 21.06.2019 17:00

Aquantity with an initial value of 3900 decays exponentially at a rate of 35% every hour. what is the value of the quantity after 0.5 days, to the nearest hundredth?

Answers: 1

Answer

Mathematics, 21.06.2019 17:40

Find the value of ax 4 ; a = 2, x = 1. select one: a. 2 b. 4 c. 1 d. 8

Answers: 2

Answer

Mathematics, 21.06.2019 19:00

You are at a restaurant and owe $38 for your meal. you want to leave a 20% tip. how much will your tip be?

Answers: 1

Answer

You know the right answer?

If sinA = 5/12 and 90° < A < 180°, and tanB = 5 and 180° < B < 270°, determine the exact...