how do you use doubel-angle formulas to find sin, cos, and tan
To use the double-angle formulas to find the values of sin, cos, and tan, you need to understand the formulas and follow a step-by-step process. Let's go through each trigonometric function:
1. Sin: The double-angle formula for sin is:
sin(2θ) = 2sin(θ)cos(θ)
To find sin(θ), given sin(2θ), you can rewrite the formula as:
sin(θ) = (1/2) * sin(2θ) / cos(θ)
First, use a calculator or lookup table to find the value of sin(2θ). Then, determine the value of cos(θ). Finally, substitute these values into the formula to find sin(θ).
2. Cos: The double-angle formula for cos is:
cos(2θ) = cos²(θ) - sin²(θ)
To find cos(θ), given cos(2θ), you can rearrange the formula as:
cos²(θ) = (cos(2θ) + 1) / 2
Then, take the square root of both sides to find cos(θ). However, note that the sign of cos(θ) cannot be determined solely from this formula. You need additional information from the quadrant in which θ lies to determine the correct sign.
3. Tan: The double-angle formula for tan is:
tan(2θ) = (2tan(θ)) / (1 - tan²(θ))
To find tan(θ), given tan(2θ), you can rearrange the formula as:
tan(θ) = tan(2θ) / (2 + tan²(θ))
Calculate the value of tan(2θ) using a calculator or lookup table. Then, substitute this value into the formula to find tan(θ).
Remember to pay attention to the restrictions of the double-angle formulas. Some formulas may not be valid for certain values of θ, such as when cos(θ) or tan(θ) is zero.
These procedures should help you use the double-angle formulas to find the values of sin, cos, and tan for a given angle, given the values for double that angle.