Given that sin theta= 3/5 and the terminal side is in quadrant II, find tan (pi/2 - theta)
make a triangle and you should reconize the 3-4-5 right-angled triangle.
but we know tan(π/2 - Ø) = cotØ , coterminal ratios
since Ø is in II , tan Ø = -3/4
cotØ = 1/tanØ = -4/3
Let's find the value of tan (pi/2 - theta).
1. First, let's find the value of pi/2 - theta. Since the terminal side is in the second quadrant and sin theta is positive, we know that theta is between 0 and pi/2. Therefore, pi/2 - theta will be positive.
pi/2 - theta = pi/2 - arcsin(3/5)
2. To simplify the expression, we need to find the value of arcsin(3/5). We know that sin theta = 3/5, so we can use the inverse sine function to find the value of theta.
arcsin(3/5) = theta
3. Now, let's substitute the value of theta into the expression from step 1.
pi/2 - theta = pi/2 - arcsin(3/5)
4. To find tan(pi/2 - theta), we need to express it in terms of sine and cosine.
tan(pi/2 - theta) = sin(pi/2 - theta) / cos(pi/2 - theta)
5. Using the trigonometric identities, we have:
sin(pi/2 - theta) = cos(theta) (identity: sin(pi/2 - theta) = cos(theta))
cos(pi/2 - theta) = sin(theta) (identity: cos(pi/2 - theta) = sin(theta))
6. Substituting these identities into the expression from step 4, we get:
tan(pi/2 - theta) = cos(theta) / sin(theta)
7. Since we know that sin(theta) = 3/5, we can substitute it into the expression from step 6.
tan(pi/2 - theta) = cos(theta) / (3/5)
8. Finally, we need to find the value of cos(theta). Since sin(theta) = 3/5 and the terminal side is in the second quadrant, we can use the Pythagorean identity to find the value of cos(theta).
cos(theta) = sqrt(1 - sin^2(theta))
= sqrt(1 - (3/5)^2)
= sqrt(1 - 9/25)
= sqrt(16/25)
= 4/5
9. Now, we can substitute the values of cos(theta) and sin(theta) into the expression from step 7.
tan(pi/2 - theta) = (4/5) / (3/5)
= (4/5) * (5/3)
= 4/3
Therefore, tan(pi/2 - theta) = 4/3.
To find the value of tan(pi/2 - theta), we need to use the trigonometric identities and the information given.
First, let's calculate the value of cos(theta). Since sin(theta) = 3/5 and the terminal side is in quadrant II, we know that sin(theta) is positive, and cos(theta) will be negative in that quadrant. Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can solve for cos(theta):
sin^2(theta) + cos^2(theta) = 1
(3/5)^2 + cos^2(theta) = 1
9/25 + cos^2(theta) = 1
cos^2(theta) = 1 - 9/25
cos^2(theta) = 16/25
cos(theta) = -4/5
Now, we can use the identity tan(pi/2 - theta) = cot(theta) = cos(theta) / sin(theta). Substituting the values we found:
tan(pi/2 - theta) = cot(theta) = (-4/5) / (3/5)
tan(pi/2 - theta) = cot(theta) = -4/3
Therefore, tan(pi/2 - theta) is equal to -4/3.