Question

Find the exact value of the expression. sin(cos^-1(3/4) − tan^-1(1/ 3)

Answer 1

To find the exact value of sin(cos⁻¹(3/4) - tan⁻¹(1/3)), construct right triangles to determine the sine and cosine of each angle, and then apply the sine angle subtraction formula.

Step-by-step explanation:

The question asks for the exact value of the trigonometric expression sin(cos⁻¹(3/4) − tan⁻¹(1/3)). To solve this, let's represent the angles cos⁻¹(3/4) and tan⁻¹(1/3) as A and B, respectively, meaning A is an angle whose cosine is 3/4 and B is an angle whose tangent is 1/3.

Create two right-angled triangles, one for each angle. For angle A, the adjacent side is 3, the hypotenuse is 4, and using the Pythagorean theorem, the opposite side is √(4² - 3²) = √7. For angle B, the opposite side is 1, the adjacent side is 3, and the hypotenuse is √(1² + 3²) = √10.

Now evaluate sin(A - B) using the angle subtraction formula for sine: sin(A - B) = sin(A)cos(B) − cos(A)sin(B). Plug in the values: sin(A) = √7/4, cos(A) = 3/4 (from the first triangle), cos(B) = 3/√10, and sin(B) = 1/√10 (from the second triangle). Therefore, sin(A - B) = (√7/4)(3/√10) − (3/4)(1/√10), which simplifies to the exact value (3√7 − 3)/(4√10).

Answer 2

The exact value of sin[cos⁻¹(3/4) - tan⁻¹(1/3)] is 39/100 or 0.39.

To find this value, we start by considering cos⁻¹(3/4), which means finding the angle whose cosine is 3/4;

cos⁻¹(3/4) = 41.41

Next, we have tan⁻¹(1/3), which is the angle whose tangent is 1/3;

tan⁻¹(1/3) = 18.43

Subtracting these angles gives;

cos⁻¹(3/4) - tan⁻¹(1/3) = 41.41 - 18.43

cos⁻¹(3/4) - tan⁻¹(1/3) = 22.98

So;

sin[cos⁻¹(3/4) - tan⁻¹(1/3)] = sin 22.98

sin[cos⁻¹(3/4) - tan⁻¹(1/3)] = 0.39

0.39 expressed as a fraction is 39/100

In conclusion, the exact value of sin[cos⁻¹(3/4) - tan⁻¹(1/3)] is 0.39 or 39/100 as a fraction, representing the sine of the difference between the angle whose cosine is 3/4 and the angle whose tangent is 1/3.