use the half angle formula to determine the exact values of sin, cos, and tan of the angle.
pie/8
and
pie/12
I will be happy to check your thinking on this.
i honestly don't know what to do or how to start anyways
the half-angle formulas let you split an angle in half
there is sin 2A = 2sinAcosA and
cos 2A = cos^2 A - sin^2 A or
cos 2A = 2cos^2 A - 1 or
cos 2A = 1 - 2sin^2 A
I will do the pi/12 angle for you.
pi/6 rads = 30 degrees, and pi/12 = 15 degrees
From the 30-60-90 triangle you should know
sin30 = 1/2 and cos30 = ?3/2
cos 30 = 1 - 2sin^2 15
2sin^2 15 = 1-?3/2
sin ^2 15 = 1/2 - ?3/4
= (2-?3)4
so sin 15 = ?(2-?3)/2
or sin pi/12 = ?(2-?3)/2
now ;use the other version of cos 2A to get cos pi/12
and since tanx = sinx/cosx, divide the two results to get the tangent.
verify your answer with your calculator
for the first pi/8 is half of pi/4 which is 45 degrees
you should know the trig rations for 45 degrees.
To determine the exact values of sin, cos, and tan of the angles π/8 and π/12 using the half-angle formula, follow these steps:
1. Start with the half-angle formula for sine: sin(x/2) = ±√[(1 - cos(x))/2].
2. Substitute the given angle into the formula, which gives sin(π/16) = ±√[(1 - cos(π/8))/2].
3. To solve for cos(π/8), square both sides of the equation: [sin(π/16)]^2 = (1 - cos(π/8))/2. Rearrange the equation to isolate cos(π/8): cos(π/8) = 1 - 2[sin(π/16)]^2.
4. Evaluate sin(π/16): sin(π/16) is not a conventional trigonometric angle, so you'll need to use a calculator or reference table to approximate its value. Let's assume it is approximately 0.196.
5. Substitute the approximate value into the equation from step 3: cos(π/8) = 1 - 2(0.196)^2. Simplify this expression to find the approximate value of cos(π/8).
6. Once you have the approximate value of cos(π/8), you can determine the value of sin(π/8) using the equation sin^2(π/8) + cos^2(π/8) = 1. Rearrange the equation to isolate sin(π/8): sin(π/8) = ±√(1 - cos^2(π/8)).
7. Substitute the approximate value of cos(π/8) into the equation from step 6 and simplify to find the approximate value of sin(π/8).
8. To find tan(π/8), use the identity tan(x) = sin(x)/cos(x). Substitute the approximate values of sin(π/8) and cos(π/8) into this equation to find the approximate value of tan(π/8).
Follow the same steps for the angle π/12 to determine the exact values of sin, cos, and tan.