Question

For cos(theta) = -3/5, and pi/2 < theta < pi, find the following trig values

sin theta
tan theta
csc theta
sec theta
cot theta

Answer 1

For the given range of theta, cos(theta) is negative, which means that theta is in the third or fourth quadrant. In these quadrants, the sine function is positive, the tangent function is negative, and the cotangent function is negative.

Since cos(theta) = -3/5, we can use the Pythagorean identity to find the value of sin(theta):

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

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

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

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

sin^2(theta) = 16/25

sin(theta) = sqrt(16/25) = 4/5

Therefore, the value of sin(theta) is 4/5.

We can also use the identity cot(theta) = 1/tan(theta) to find the value of cot(theta):

cot(theta) = 1/tan(theta)

cot(theta) = 1/(sin(theta)/cos(theta))

cot(theta) = cos(theta)/sin(theta)

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

cot(theta) = -3/4

Therefore, the value of cot(theta) is -3/4.

The values of the other trigonometric functions can be found using the definitions of these functions:

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

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

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

Therefore, the values of the trigonometric functions for the given range of theta are:

sin(theta) = 4/5

tan(theta) = -4/3

csc(theta) = 5/4

sec(theta) = -5/3

cot(theta) = -3/4

Answer 2

Given that cos(theta) = -3/5 and theta is in the second quadrant (pi/2 < theta < pi), we know that sin(theta) will be positive because in the second quadrant sine is positive.

We start by finding sin(theta) using the Pythagorean Identity for sin and cos, which is sin^2(theta) + cos^2(theta) = 1.

Solving the Pythagorean Identity equation for sin gives us sin^2(theta) = 1 - cos^2(theta). Using the given cos(theta) and solving for sin(theta), we find that
sin(theta) = sqrt[1 - (-3/5)^2] = 0.8.

Now, tan(theta) is defined as the ratio sin(theta) / cos(theta). Using the given cos(theta) and the calculated sin(theta), we find that
tan(theta) = sin(theta) / cos(theta) = 0.8 / -0.6 = -1.33 (rounded to 2 decimal places).

Using the definitions of the other trigonometric ratios csc, sec and cot, we find that

The cosecant function, csc(theta), is the reciprocal of sin(theta). Therefore,
csc(theta) = 1 / sin(theta) = 1 / 0.8 = 1.25

The secant function, sec(theta), is the reciprocal of cos(theta). Therefore,
sec(theta) = 1 / cos(theta) = 1 / -3/5 = -1.67 (rounded to 2 decimal places).

The cotangent function, cot(theta), is the reciprocal of tan(theta). Therefore,
cot(theta) = 1 / tan(theta) = 1 / -1.33 = -0.75 (rounded to 2 decimal places).

So, for theta in the second quadrant with cos(theta) = -3/5, we find that sin(theta) = 0.8, tan(theta) = -1.33, csc(theta) = 1.25, sec(theta) = -1.67, and cot(theta) = -0.75.