Finding remaining trigonometric functions of theta

[rachelmaddie]

Hi. I need my work checked please.

Since Angle θ is in Quadrant I, all the coordinates are positive.

To find the value of cos(θ), we have sin(θ) = 84/85
We know that sin^2(θ) + cos^2(θ) = 1
Substitute,
(84/85)^2 + cos^2(θ) = 1
(7,056/7,225) + cos^2(θ) = 1
cos^2(θ) = 1 - (7,056/7,225)
cos^2(θ) = (169/7,225)
cos(θ) = (13/85)
cos(θ) = (13/85)

To find the value of tan(θ), we have tan(θ) = sin(θ)/cos(θ)
We know that sin(θ) = 84/85 and cos(θ) = (13/85)
Substitute,
tan(θ) = (84/85)/(13/85)
tan(θ) = 84/13
tan(θ) = 84/13

To find the value of cot(θ), we have cot(θ) = 1/tan(θ)
We know that tan(θ) = 84/13
Therefore, cot(θ) = 13/84
cot(θ) = 13/84

To find the value of sec(θ), we have sec(θ) = 1/cos(θ)
We know that cos(θ) = (13/85)
Therefore, sec(θ) = 85/13
sec(θ) = 85/13

To find the value of csc(θ), we have csc(θ) = 1/sin(θ)
We know that sin(θ) = 84/85
Therefore, csc(θ) = 85/84
csc(θ) = 85/84

 

[LCKurtz]

All correct.

 

[HallsofIvy]

Here's another way to do that. Since \(\displaystyle sin(x)= \frac{84}{85}\) we can model this as a right triangle with "opposite side" of length 84 and hypotenuse of length 85. By the
"Pythagorean theorem" the "near side" has length \(\displaystyle \sqrt{85^2- 84^2}= \sqrt{169}= 13\). You can calculate all other trig functions by:
cos(x)= "near side"/hypotenuse= 13/85
tan(x)= "opposite side"/"near side"= 13/84
cot(x)= "near side"/"opposite side"= 84/13
sec(x)= hypotenuse/"near side"= 85/13
csc(x)= hypotenuse/"opposite side"= 85/84