Evaluate (exact answers):
a) sin^-1(cos30)
Need help. What do I do?
b) sin[(cos^-1((sqrt2)/(2)))+[(sin^-1((sqrt2)/(2)))]
= sin(45+45)
= sin90
= 1
Is this correct?
c) cos(arctan5/7)
= 7/(sqrt74)
= (7sqrt74)/74
Is this correct?
Please and Thank you!
To evaluate the expressions, we can use the inverse trigonometric functions and the trigonometric identities. I'll explain how to solve each problem step-by-step:
a) To evaluate sin^(-1)(cos30), first, we need to find the value of cos30. The cosine of 30 degrees can be found using the unit circle or a calculator, and it equals (√3)/2. Now we have sin^(-1)((√3)/2). This means we are looking for an angle whose sine is (√3)/2. The angle that satisfies this condition is 60 degrees (π/3 radians). Therefore, sin^(-1)((√3)/2) = 60 degrees or π/3 radians.
b) Let's break down sin[(cos^(-1)((√2)/2)) + (sin^(-1)((√2)/2))].
First, we evaluate cos^(-1)((√2)/2), which means we are looking for an angle whose cosine is (√2)/2. This angle is 45 degrees (π/4 radians). So, we substitute this value into the expression: sin[(π/4) + (sin^(-1)((√2)/2))].
Now, we evaluate sin^(-1)((√2)/2), which means we are looking for an angle whose sine is (√2)/2. This angle is also 45 degrees (π/4 radians). Substituting this value into the expression: sin[(π/4) + (π/4)].
Adding the angles gives us sin(π/2) = 1.
Therefore, your answer for b) is correct: sin[(cos^(-1)((√2)/2)) + (sin^(-1)((√2)/2))] = 1.
c) To evaluate cos(arctan(5/7)), we start by finding arctan(5/7), which means we need to find an angle whose tangent is 5/7. Let's call this angle θ. Using the right triangle trigonometry, we have:
tan(θ) = 5/7.
This implies that the opposite side is 5 and the adjacent side is 7 in the right triangle. Now, we can use the Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 5^2 + 7^2
hypotenuse = √74.
Now, we can evaluate cos(arctan(5/7)). Recall that cosine is the ratio of the adjacent side to the hypotenuse in a right triangle. Substituting the values we found:
cos(arctan(5/7)) = adjacent/hypotenuse = 7/√74.
To rationalize the denominator, multiply both the numerator and denominator by √74:
cos(arctan(5/7)) = (7/√74) * (√74/√74) = (7√74)/74.
Therefore, your answer for c) is correct: cos(arctan(5/7)) = (7√74)/74.
I hope this explanation helps! Let me know if you have any more questions.