Evaluate the following. Show you work:
A). tan^-1(tan(5π/6))
B). cos^-1(cos(-π/4))
Can someone please show me the steps to evaluating this problem. I've been working this for 2 hours today and all of last night, I still can't seem to figure it out. Please leave notes if possible.
each inverse trig function has a principal domain. Review those.
tan 5π/6 = -tan π/6 = -1/√3
but, tan^-1(-1/√3) = -π/6
cos -π/4 = cos π/4 = 1/√2
but, cos^-1(1/√2) = π/4
Thank you.
To evaluate the given expressions, we need to understand the trigonometric ratios and their inverses. Let's break down the steps for each expression:
A). tan^-1(tan(5π/6)):
1. Start by finding the value of the innermost function, tan(5π/6).
- We know that tan(θ) = sin(θ)/cos(θ), so tan(5π/6) = sin(5π/6)/cos(5π/6).
2. Evaluate sin(5π/6) and cos(5π/6):
- First, we need to determine the reference angle for 5π/6. Since it falls in the third quadrant (where cos and sin values are negative), we can find the reference angle by subtracting it from 2π (one full revolution).
- Reference angle = 2π - 5π/6 = 12π/6 - 5π/6 = 7π/6.
- Now, we can use the reference angle to find the values of sin and cos:
- sin(5π/6) = sin(7π/6) = sin(π/6) (since sine function is periodic every 2π)
= 1/2.
- cos(5π/6) = cos(7π/6) = cos(π/6)
= sqrt(3)/2.
3. Substitute the calculated values into tan(5π/6) = sin(5π/6)/cos(5π/6):
- tan(5π/6) = (1/2) / (sqrt(3)/2)
= 1/sqrt(3) (Multiply the numerator and denominator by 2 to simplify).
4. Finally, evaluate tan^-1(1/sqrt(3)):
- tan^-1(1/sqrt(3)) is asking "What angle has a tangent of 1/sqrt(3)?"
- We find this angle through the inverse tangent function, which gives us the angle whose tangent is the given value.
- Using a calculator, the angle is approximately 30 degrees or π/6 radians.
So, the evaluation of tan^-1(tan(5π/6)) is approximately 30 degrees or π/6 radians.
B). cos^-1(cos(-π/4)):
1. Begin by determining the value of the innermost function, cos(-π/4).
- The cosine function is an even function, so cos(-θ) = cos(θ).
- Therefore, cos(-π/4) = cos(π/4).
2. Evaluate cos(π/4):
- We know that cos(π/4) = sqrt(2)/2. This is a common value from the unit circle.
3. Substitute the value into cos^-1(cos(-π/4)):
- cos^-1(cos(-π/4)) is asking "What angle has a cosine of sqrt(2)/2?"
- We find this angle through the inverse cosine function, which gives us the angle whose cosine is the given value.
- Using a calculator, the angle is approximately 45 degrees or π/4 radians.
Therefore, the evaluation of cos^-1(cos(-π/4)) is approximately 45 degrees or π/4 radians.
Note: When working with inverse trigonometric functions, it's important to ensure the accuracy of calculators and units being used (degrees/radians).