Given cosθ = 3/5, find the five other trigonometric function values. 15

[tex]cos(\theta )=\cfrac{\stackrel{adjacent}{3}}{\underset{hypotenuse}{5}}\qquad \impliedby \textit{let's now find the \underline{opposite side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \sqrt{5^2-3^2}=b\implies \sqrt{25-9}=b\implies \sqrt{16}=b\implies 4=b \\\\[-0.35em] ~\dotfill[/tex]

[tex]sin(\theta )=\cfrac{\stackrel{opposite}{4}}{\underset{hypotenuse}{5}}\qquad \qquad tan(\theta )=\cfrac{\stackrel{opposite}{4}}{\underset{adjacent}{3}}\qquad \qquad cot=\cfrac{\stackrel{adjacent}{3}}{\underset{opposite}{4}} \\\\\\ sec(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{adjacent}{3}}\qquad \qquad csc(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{opposite}{4}}[/tex]

The values of the five other trigonometric functions value for the same input θ for which cos(θ) = 3/5 are:

Trigonometric ratios for a right angled triangle are from the perspective of a particular non-right angle.

In a right angled triangle, two such angles are there which are not right angled(not of 90 degrees).

The slant side is called hypotenuse.

From the considered angle, the side opposite to it is called perpendicular, and the remaining side will be called base.

From that angle (suppose its measure is θ),

[tex]\sin(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of Hypotenuse}}\\\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}\\\\\tan(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of base}}\\\\\cot(\theta) = \dfrac{\text{Length of base}}{\text{Length of perpendicular}}\\\\\sec(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of base}}\\\\\csc(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of perpendicular}}\\[/tex]

If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:

[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]

where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).

We're given that:

cosθ = 3/5 for some angle θ

Since we've got:

[tex]\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}[/tex]

Therefore, we have:

[tex]\dfrac{3}{5} = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}[/tex]

Let we consider a right angled triangle in which there is hypotenuse of length 5 units and base (from the perspective of one of its non-right angle) of 3 units (as shown in the image attached below).

(we couldve taken 3x and 5x instead of 3 and 5, for any positive real number value of 'x' as when we would take their ratio, that common factor 'x' would get cancelled out. We can think of 3 and 5 as the special case of 3x and 5x when x = 1)

Then, from that perspective, let the perpendicular be of the length 'p' units, then as per the pythagoras theorem, we get:

[tex]p^2 + 3^2 = 5^2\\p = \sqrt{25 - 9} = \sqrt{16} = 4 \: \rm units[/tex]

(took only the positive root to remove the square term because the value of p denotes length, which is a non-negative quantity).

Thus, we have:

From the perspective of the angle θ:

Thus, using these values, and the definition of the trigonometric ratios, we get:

Thus, the values of the five other trigonometric functions value for the same input θ for which cos(θ) = 3/5 are:

Learn more about Pythagoras theorem here:

https://brainly.com/question/12105522

Learn more about trigonometric ratios here:

https://brainly.com/question/22599614

#SPJ4

please help me will give branlist

can some help me out with this EXTRA POINTS!!

You use 4 gallons of water on 30 plants in your garden. At that rate, how much water
will it take to water 45 plants?

Solve the equation:
4x-x=600

Write down a number which is •a multiple of 4. •a factor of 400. •not 4 or 400

Arnav is also selling boxes of cookies he sells each box for 3.75 his goal to make more than 30 how many boxes of cookies could he sell to reach his goal

please help me out with these two problems

find X and Y

what are the coordinates of a point where graph of y=x2+3x+2 crosses the x axis

Help, me and my partner are timed, no links thx

please HELPPP! I have been stuck on this for too long.

what is the nature of triangle formed by the lines x+y=0 , 3x+y=0 , x+3y-4=0

For each equation how many solutions why?
Question e

Help PLSS
(will give brainliest!)

Phil bought a new calculator. It cost $12.40. The tax rate is 7.25%. Calculate the sales tax.

Name the circle
Will give a lot of points!!! ASAP

How many apples could you buy if you went to the store with five dollars? *

You are starting a savings account for college. You put $1,000 in as your starting balance. You earn simple interest at 10% every year. You also must pay 30% income tax on the interest earned annually. Calculate the balance for the account after 5 years

given right triangle ABC with altitude BD drawn to hypotenuse AC. If BD = 24 nd DC = 12, what is the length of AD?

An office supply store in Newport purchased a printer at a cost of $175 and marked it up 45%. Later on, Oliver bought the printer. If the sales tax in Newport is 12%, how much did he pay in all?
please help asap would appreciate it ty

Ben runs 3/4 mile each day.How many Miles Does ben run in 12 days

What is the correct answer?

log2^2-log2^16 what is the answer

A square pyramid has
a base edge of 8 inches
and a surface area of 304
square inches. Find the
Slant height of the
pyramid.

Please Help! will give Brainlest!

Andrea thought that x = 15 would be a good guess to solve 6x + 19 = 150. Should her second guess be higher or lower? Why?

the drop down box numbers are 39, 90, 120, and 180

At the state fair, 2 bags of mini donuts and 4 turkey legs cost $18.50. for $13.75, you can buy 3 bags of mini donuts and 2 turkey legs.using these amounts, which system of equations can be used to find the cost of 1 bag of mini donuts?

PLEASE PLEASE HELP !! The translation below is used on all three triangles.
(x, y) → (x, y + 1)
What happens to the triangles? Which answer would it be A, B, C, or D?
(I WILL MARK AS BRAINLIEST! PLEASE HELP!)

(A) They all move 1 unit to the right.
(B)They all move 1 unit to the left.
(C)They all move 1 unit down.
(D)They all move 1 unit up.

what is the numerical coefficient of a+4a^3-6a^2+2a^4

Write the expression as a product.
25-(a + 7)^2