1. sin(tan^-1(4/3))=4/5

2. find the angle whose tangent equals 0.
tan^-1(cos(pi/2))=0

both are correct, but I think the answer to #2 does not relate well to the question.

I'd have said tan^-1(0) = 0 since tan(0) = 0

courtney has a three digit cards.

they are 6 8 and 3.

what is the largest number that courtney can write using these digit cards?

To solve these trigonometric equations, we will use properties of inverse trigonometric functions and some basic trigonometry identities.

1. To solve the equation sin(tan^(-1)(4/3)) = 4/5:

- First, let's consider the expression inside the sine function, which is tan^(-1)(4/3). This represents the angle whose tangent is 4/3.
- Using the inverse tangent function, we can find this angle: tan^(-1)(4/3) ≈ 53.13°.
- Now, we substitute this angle into the sine function: sin(53.13°).
- To find the value of sin(53.13°), we can use the unit circle or a calculator. In this case, sin(53.13°) ≈ 0.8.
- Finally, we compare this value to 4/5. Since sin(tan^(-1)(4/3)) = 0.8 and 4/5 = 0.8, the equation is true.

2. To find the angle whose tangent equals 0:

- We know that the tangent function relates the ratio of the opposite side to the adjacent side of a right triangle.
- If the tangent is 0, it means that the opposite side is 0, and therefore, the angle is 0° or any multiple of 180°.
- We can use the inverse tangent function to find the angle: tan^(-1)(cos(pi/2)).
- Substitute the value into the inverse tangent function: tan^(-1)(0).
- The inverse tangent of 0 is 0°, so the angle whose tangent equals 0 is 0°.

Remember to verify the second problem by substituting the angle back into the tangent function to ensure it equals 0.