Find the exact value of each of the other 5 trigonometric functions for the angle of theta (without finding theta) when cos theta is 3/5
To find the exact values of the other trigonometric functions (sine, tangent, cosecant, secant, and cotangent) given that cos(theta) = 3/5, we can use the Pythagorean identity and the definitions of the trigonometric functions.
1. Start by finding the value of sin(theta):
- Since cos(theta) = 3/5, we can use the Pythagorean identity: cos^2(theta) + sin^2(theta) = 1.
- Plugging in cos(theta) = 3/5, we get:
(3/5)^2 + sin^2(theta) = 1
9/25 + sin^2(theta) = 1
sin^2(theta) = 1 - 9/25
sin^2(theta) = 25/25 - 9/25
sin^2(theta) = 16/25
- Taking the square root of both sides gives:
sin(theta) = √(16/25)
- Simplifying, we have:
sin(theta) = 4/5
2. Next, find the value of tan(theta):
- The tangent function is defined as tan(theta) = sin(theta)/cos(theta), so we can substitute the values we just found:
tan(theta) = (4/5) / (3/5)
= 4/5 * 5/3
= 4/3
3. Moving on to the cosecant:
- The cosecant function is the reciprocal of the sine function, so we can invert the value of sin(theta):
csc(theta) = 1 / sin(theta)
= 1 / (4/5)
= 5/4
4. Finding the secant:
- The secant function is the reciprocal of the cosine function, so we can invert the value of cos(theta):
sec(theta) = 1 / cos(theta)
= 1 / (3/5)
= 5/3
5. Finally, determining the cotangent:
- The cotangent function is the reciprocal of the tangent function, so we can invert the value of tan(theta):
cot(theta) = 1 / tan(theta)
= 1 / (4/3)
= 3/4
In summary, for the given value of cos(theta) = 3/5, the exact values of the other trigonometric functions are:
sin(theta) = 4/5
tan(theta) = 4/3
csc(theta) = 5/4
sec(theta) = 5/3
cot(theta) = 3/4