For positive acute angles AA and B,B, it is known that \tan A=\frac{28}{45}tanA= 45 28 ​ and \cos B=\frac{3}{5}.CosB= 5 3 ​ . Find the value of \cos(A+B)cos(A+B) in simplest form.

Answer:

[tex]\cos(A + B) = \frac{23}{265}[/tex]

Step-by-step explanation:

Given

[tex]\tan A=\frac{28}{45}[/tex]

[tex]\cos B=\frac{3}{5}[/tex]

Required

[tex]\cos(A+B)[/tex]

In trigonometry:

[tex]\cos(A+B) = \cos\ A\ cosB - sinA\ sinB[/tex]

We need to solve for cosA, sinA and sinB

Given that:

[tex]\tan A=\frac{28}{45}[/tex] and the tangent of an angle is a ratio of the opposite side (x) to the adjacent side (y),

So, we have:

[tex]x =28[/tex] and [tex]y = 45[/tex]

For this angle A, we need t calculate its hypotenuse (z).

Using Pythagoras,

[tex]z^2 = x^2 + y^2[/tex]

[tex]z = \sqrt{x^2 + y^2[/tex]

[tex]z = \sqrt{28^2 + 45^2[/tex]

[tex]z = \sqrt{2809[/tex]

[tex]z = 53[/tex]

From here, we can calculate sin and cos A

[tex]\sin A= \frac{opposite}{hypotenuse}[/tex] and [tex]\cos A= \frac{adjacent}{hypotenuse}[/tex]

[tex]\sin A= \frac{x}{z}[/tex]

[tex]\sin A= \frac{28}{53}[/tex]

[tex]\cos A= \frac{y}{z}[/tex]

[tex]\cos A= \frac{45}{53}[/tex]

To angle B.

Give that

[tex]\cos B=\frac{3}{5}[/tex] and the cosine of an angle is a ratio of the adjacent side (a) to the hypotenuse side (c),

So, we have:

[tex]a = 3[/tex] and [tex]b = 5[/tex]

For this angle B, we need to calculate its opposite (b).

Using Pythagoras,

[tex]c^2 = a^2 + b^2[/tex]

[tex]5^2 = 3^2 + b^2[/tex]

[tex]25 = 9 + b^2[/tex]

Collect like terms

[tex]b^2 = 25 - 9[/tex]

[tex]b^2 = 16[/tex]

[tex]b = \sqrt{16[/tex]

[tex]b = 4[/tex]

From here, we can calculate sin B

[tex]\sin B= \frac{opposite}{hypotenuse}[/tex]

[tex]\sin B = \frac{b}{c}[/tex]

[tex]\sin B = \frac{4}{5}[/tex]

Recall that:

[tex]\cos(A+B) = \cos\ A\ cosB - sinA\ sinB[/tex]

and

[tex]\cos B=\frac{3}{5}[/tex]      [tex]\sin B = \frac{4}{5}[/tex]       [tex]\sin A= \frac{28}{53}[/tex]      [tex]\cos A= \frac{45}{53}[/tex]

[tex]\cos(A + B) = \frac{45}{53} * \frac{3}{5} - \frac{28}{53} * \frac{4}{5}[/tex]

[tex]\cos(A + B) = \frac{135}{265} - \frac{112}{265}[/tex]

[tex]\cos(A + B) = \frac{135-112}{265}[/tex]

[tex]\cos(A + B) = \frac{23}{265}[/tex]

The value of [tex]\cos (A+B)[/tex] is [tex]-\frac{96}{265}[/tex].

Trigonometrically speaking, the sine, cosine and tangent of an angle are described by the following formulas:

[tex]\sin \theta = \frac{y}{\sqrt{x^{2}+y^{2}}}[/tex] (1)

[tex]\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}[/tex] (2)

[tex]\tan \theta = \frac{y}{x}[/tex] (3)

Where:

And we have the following trigonometric identity for the cosine of the sum of angles:

[tex]\cos (A+B) = \cos A\cdot \cos B - \sin A \cdot \sin B[/tex] (4)

By (1) and (2), we rewrite (4):

[tex]\cos (A+B) = \frac{x_{A}\cdot x_{B}}{\sqrt{x_{A}^{2}+y_{A}^{2}}\cdot \sqrt{x_{B}^{2}+y_{B}^{2}}} - \frac{y_{A}\cdot y_{B}}{\sqrt{x_{A}^{2}+y_{A}^{2}}\cdot \sqrt{x_{B}^{2}+y_{B}^{2}}}[/tex]

If we know that [tex]x_{A} = 28[/tex], [tex]y_{A} = 45[/tex], [tex]x_{B} = 3[/tex] and [tex]y_{B} = 4[/tex], then the cosine for the sum of angles is:

[tex]\cos (A+B) = \frac{28\cdot 3}{\sqrt{28^{2}+45^{2}}\cdot \sqrt{3^{2}+4^{2}}}-\frac{45\cdot 4}{\sqrt{28^{2}+45^{2}}\cdot \sqrt{3^{2}+4^{2}}}[/tex]

[tex]\cos (A+B) = -\frac{96}{265}[/tex]

The value of [tex]\cos (A+B)[/tex] is [tex]-\frac{96}{265}[/tex].

We kindly invite to check this question on trigonometric identities: https://brainly.com/question/24836845

PLEASE HELP! Will give brainliest
5. What is the next term of the sequence -3, 6, -12, 24,... , and what it's arithmetic sequence?

In each bouquet of flowers, there are 5 roses and 8 white carnations. Complete the table to find
many roses and carnations there are in 5 bouquets of flowers.




There are (____) roses and (____) carnations in 5 bouquets

Which equation demonstrates the distributive property? A = 7(1+6)=7(6+1) B = (7x1) x 6 = 7(1x6) C = (7x1) x 6 = 7(6x1) D = 7(1+6) = (7x1) + (7x6) HELP END OF SEMSSTER

Question 5 (1 point)
Supplementary angles always
оа
b
equivalent.
add up to 180 degrees.
add up to 360 degrees.
add up to 90 degrees.
ос
Od

find the slope of the graph​

Kąty α i β są kątami przyległymi. Kąt wyznaczony przez dwusieczne kątów α oraz β ma miarę A) 90∘ B) 4 5∘ C) 60∘ D) różną, w zależności od miar kątów α i β

help me on this question

Can someone help me!!

-11/12 minus
3/8
Select one:
A.
-1 7/24
B.
-8/24
C.
8/4
D
-13/24

Please please help and show how you got the answer! I WILL MARK YOU AS BRAINLIEST

Two kinds of tickets to an outdoor concert were sold: lawn tickets and seat tickets. Less than 400 tickets in total were sold. Lawn tickets cost $30 each and seat tickets cost $50 each. The organizers want to make at least $14,000 from ticket sales.

Write a system of inequalities to represent the situation. Be sure to define your variables.

Pls explain how to solve this problem pls ASAP if you know it pls!

what is 6÷102??
for math ​

2x* + 8x8 – 25x² - 6x +14 ?
Which expression is equivalent to
x +6
(1) 2x3 + 4x² + x - 12 +
86
r +6
(2) 2x3 – 4x2
-x + 14
(3) 2x3 - 4x2 - r +
14
1 +6
(4) 2x² - 4x² – x
Please help

Which statement best describes the effect of replacing the function f(x) = 2x − 2 with the function g(x) = 2x + 3? (1 point)

The graph shifts 1 units left.
The graph shifts 5 units left.
The graph shifts 1 unit right.
The graph shifts 3 units right.

A high school basketball team scored 60 points in last week’s game. The team scored a total of 27 baskets; some were two-point shots and some were three-point shots. How many two-point shots did they make? How many three-point shots did they make?


x + y = 27,


2x + 3y = 60



What is the solution of the system of equations, and what does it represent?


options:

(6, 21); 6 two-point shots and 21 three-point shots

(6, 21); 6 three-point shots and 21 two-point shots

(21, 6); 21 two-point shots and 6 three-point shots

(21, 6); 21 three-point shots and 6 two-point shots

Gina has 7/8 pound of nuts that she wants to divide equally into 4 bags. How can Gina determine the amount of nuts she can put in each bag? For this problem Gina must divide a 1._________ by a 2. ________. The amount of nuts in each bag will be 3._________ 7/8 because 7/8 is 4._________ 4. Gina will be able to put 5._________ pound(s) of nuts in each bag. Word Bank 1. Fraction or Whole Number 2. Fraction or Whole Number 3. Less Than or Greater Than 4. Divided By or Multiplyed By 5. 7/2, 32/7, 7/2, 7/32

Javier simplified the expression below. Find and
describe the three mistakes he made.
20x
- (203) = (5x®]* _ 5x * to
(5x®)» (20x)} – 20.x* *4

What is the solution of the equation x + 12.23 = 28.659?
X =

On January 1, 2014, the federal minimum wage was $7.25 per hour. Which graph has a slope that best represents this rate? Will give brainliest!!

if m<1 = 3x - 3 and m<5 = 7x + 23, for what value of x is l||m

One week the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1,240. The next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1,027. The following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2,091. Find the cost of a trumpet, clarinet, and violin.

Answers:
Trumpet:
Clarinet:
Violin:

Find all values of x that make the equation true.
3/x-4 = x-5/x

The cost price of a washing machine is $15,480 if the selling priceis $17,000 what is the profit or loss

The Rodriguez family's ate 1/3 of a pan of brownies, they will take One part to their block party and the other to share with the co-workers. How much of the whole pan of brownies will each plate get?

Carson is checking his work with Katie and finds that they disagree on a problem. Here’s what Katie wrote:

3x + 3 ≤ −2x + 5

3x ≤ −2x + 2

x ≤ 2



Is she right? Explain.

URGENT LOTS OF POINTS

pls help me with this question.

Number 17 please help?
Pls

plzzzz some one help I want step by step I'm confused What's 5⁄6 of £300