Evaluate tan (inverse cosine of 2/3).
a. square root of 5 divided by 2
b. square root of 5 divided by 3
c. square root of 5 divided by 5
I have a feeling that the answer would be a but what i am confused about is how do we figure out what the value is for inverse cosine of 2/3.
draw the triangle with sides side2 and hypotensue 3. That makes the opposite side sqrt5.
Then the tan of that angle is (sqrt5)/2
Thank you so much! I get it now!!!
To find the value of the inverse cosine of 2/3, we can use the definition of the inverse cosine function as follows:
cos^(-1)(2/3)
We know that the cosine function is the ratio of the adjacent side to the hypotenuse in a right triangle. Let's assume we have a right triangle with an angle X and a side length of 2 as the adjacent side and a side length of 3 as the hypotenuse.
Using the Pythagorean theorem, we can find the length of the opposite side as follows:
opposite side = √(hypotenuse^2 - adjacent side^2)
= √(3^2 - 2^2)
= √(9 - 4)
= √5
Now, we know that the sine of angle X is the ratio of the opposite side to the hypotenuse. So:
sin(X) = opposite side ÷ hypotenuse
= √5 ÷ 3
To find the value of the angle X, we can take the inverse sine:
X = sin^(-1)(√5 ÷ 3)
Therefore, the value of the inverse cosine of 2/3 is X. So, to evaluate tan(X), we can use the trigonometric identity:
tan(X) = sin(X) ÷ cos(X)
Substituting the value of sin(X) and cos(X) we found earlier:
tan(X) = (√5 ÷ 3) ÷ (2 ÷ 3)
= (√5 ÷ 3) × (3 ÷ 2)
= √5 ÷ 2
Hence, the answer is option a: square root of 5 divided by 2.