give the exact real number value without using a calculator

sin[(cos^-1)(1/4)]

sin[(tan^-1)(3/4)+(tan^-1)(12/5)]

On each of these, draw the triangle. On the first, the adjacent side is 1, the hyp is 4, so the opposite side is sqrt(16-1) So sine of that angle is sqrt15/4

Same idea on the next.

To find the exact real number value of sin[(cos^-1)(1/4)], we can follow these steps:

Step 1: Start with the innermost function, cos^-1(1/4), which represents the arccosine of 1/4.

The arccosine function gives us an angle whose cosine is 1/4. To find this angle, we use the identity cos(a) = 1/4.

Step 2: Solve for the angle.

cos(a) = 1/4

To solve for the angle a, we take the inverse cosine of both sides:

a = cos^-1(1/4)

Step 3: Evaluate cos^-1(1/4).

To evaluate this, we need to find the angle whose cosine is 1/4. This angle lies between 0 and π.

Using a calculator, we find that cos^-1(1/4) is approximately 1.3181 radians or about 75.5225 degrees.

Step 4: Now, we substitute the value of the angle (a) into the outer function sin(a).

sin[(cos^-1)(1/4)] = sin(1.3181)

Using a calculator, we find that sin(1.3181) is approximately 0.9492.

Therefore, the exact real number value of sin[(cos^-1)(1/4)] is approximately 0.9492.

Moving on to the next expression, sin[(tan^-1)(3/4) + (tan^-1)(12/5)], we can follow a similar approach:

Step 1: Start with the innermost function, tan^-1(3/4), which represents the arctangent of 3/4.

The arctangent function gives us an angle whose tangent is 3/4. To find the angle, we use the identity tan(b) = 3/4.

Step 2: Solve for the angle.

tan(b) = 3/4

To solve for the angle b, we take the inverse tangent of both sides:

b = tan^-1(3/4)

Step 3: Evaluate tan^-1(3/4).

Using a calculator, we find that tan^-1(3/4) is approximately 0.6435 radians or about 36.8699 degrees.

Step 4: Repeat Steps 1-3 for the second innermost function, tan^-1(12/5), which represents the arctangent of 12/5.

Using a calculator, we find that tan^-1(12/5) is approximately 1.1760 radians or about 67.3801 degrees.

Step 5: Now, we substitute the values of the angles (b) into the outer function sin(b1 + b2).

sin[(tan^-1)(3/4) + (tan^-1)(12/5)] = sin(0.6435 + 1.1760)

Using a calculator, we find that sin(0.6435 + 1.1760) is approximately 0.9179.

Therefore, the exact real number value of sin[(tan^-1)(3/4) + (tan^-1)(12/5)] is approximately 0.9179.