192k views
2 votes
For cos=-3/5, and pi/2< theta < pi find the following trig values.

sin
tan
csc
sec
cot

User Sleexed
by
8.5k points

1 Answer

2 votes
Sure thing, Erin! Let's find the trigonometric values for the given cosine value of -3/5.

First, let's find the sine value. Since the cosine is negative and theta is in the second quadrant (pi/2 < theta < pi), we know that the sine will be positive. We can use the Pythagorean identity to find the sine:

sin^2(theta) + cos^2(theta) = 1

Substituting the given value of cosine:

sin^2(theta) + (-3/5)^2 = 1

Simplifying:

sin^2(theta) + 9/25 = 1

sin^2(theta) = 1 - 9/25

sin^2(theta) = 16/25

Taking the square root of both sides:

sin(theta) = ±4/5

Since theta is in the second quadrant, the sine will be positive. So, the sine value is 4/5.

Next, let's find the tangent value. We know that tangent is equal to sine divided by cosine:

tan(theta) = sin(theta) / cos(theta)

Substituting the values we found:

tan(theta) = (4/5) / (-3/5)

Simplifying:

tan(theta) = -4/3

For the cosecant value, we know that it is the reciprocal of the sine:

csc(theta) = 1 / sin(theta)

Substituting the sine value we found:

csc(theta) = 1 / (4/5)

Simplifying:

csc(theta) = 5/4

For the secant value, we know that it is the reciprocal of the cosine:

sec(theta) = 1 / cos(theta)

Substituting the given cosine value:

sec(theta) = 1 / (-3/5)

Simplifying:

sec(theta) = -5/3

Finally, for the cotangent value, we know that it is the reciprocal of the tangent:

cot(theta) = 1 / tan(theta)

Substituting the tangent value we found:

cot(theta) = 1 / (-4/3)

Simplifying:

cot(theta) = -3/4

So, the trigonometric values for the given cosine value of -3/5, where pi/2 < theta < pi, are:

sin(theta) = 4/5
tan(theta) = -4/3
csc(theta) = 5/4
User Pravin
by
7.8k points