if tan 0= -1/3 and sin 0 > 0, find the exact values of the other five trigonometric functions of 0

Question

if tan 0= -1/3 and sin 0 > 0, find the exact values of the other five trigonometric functions of 0

Answer 1

Lets solve this shall we ?

So, we must find the other five trig functions given that tan0 = -1/3 and sin0 must be strictly greater than 0 such that,

cot0 is the reciprocal of tan0 which gives us cot0 = -3. Then, the Pythagorean Theorem states that

(-1)² + (3)² = x²

1 + 9 = x²

x = √(10)

Then, sin0 = - √(10) / 10. However, since sin0 must be strictly greater than 0, sin0 = + √(10) / 10. Then, cos0 = 3 / √(10) = 3√(10) / 10 after rationalizing the denominator.

sec 0 is the reciprocal of cos0 which gives sec0 = 10 / 3√(10) = 10√(10) / 30 = √(10) / 3 also after rationalizing the denominator. Finally, we finish off with finding csc0 which is the reciprocal of sin0 such that

csc0 = 10 / √(10) = 10√(10) / 10 = √(10) after rationalizing the denominator!

I hope this helped!