I need help trying to solve trigonometric identity problems
I just need help if someone could break it down step by step how to solve for:
if cos of theta is -3/5 find csc of theta
CosTheta=-3/5
so you know the adjacent side to Theta is -3, and the hypotensue is 5.
That makes the opposite side =-4,or +4
so, sinTheta=4/5 or -4/5
secTheta=-5/3
cscTheta=1/sin= 5/4 or -5/4
TanTheta=4/3 or -4/3
Theta is in quadrante II or III
Sure! To solve for csc(theta), the reciprocal of sin(theta), we can follow these steps:
Step 1: Start with the given information, which is cos(theta) = -3/5.
Step 2: Use the Pythagorean identity to find sin(theta). The Pythagorean identity states that sin^2(theta) + cos^2(theta) = 1.
Since we have cos(theta) = -3/5, we can substitute it into the formula:
sin^2(theta) + (-3/5)^2 = 1
Step 3: Simplify the equation using algebraic operations.
sin^2(theta) + 9/25 = 1
Step 4: Move the constant term to the other side of the equation.
sin^2(theta) = 1 - 9/25
sin^2(theta) = 25/25 - 9/25
sin^2(theta) = 16/25
Step 5: Take the square root of both sides of the equation to solve for sin(theta).
sin(theta) = ± √(16/25)
Step 6: Simplify the square root.
sin(theta) = ± (4/5)
Step 7: Finally, find csc(theta) by taking the reciprocal of sin(theta).
csc(theta) = 1/sin(theta)
csc(theta) = 1/(±4/5)
csc(theta) = ±5/4
Thus, csc(theta) can be either 5/4 or -5/4, depending on the positive or negative sign of sin(theta).