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Math  /  Trigonometry

Questionsin⁡−1(1.94)=\sin ^{-1}(1.94)= □\square tan⁡−1(2.27)=\tan ^{-1}(2.27)= □\square cos⁡−1(−0.62)=\cos ^{-1}(-0.62)= □\square

Studdy Solution

STEP 1

1. The function sin⁡−1(x)\sin^{-1}(x) is the inverse sine function, also known as arcsine, which is defined for −1≤x≤1-1 \leq x \leq 1.
2. The function tan⁡−1(x)\tan^{-1}(x) is the inverse tangent function, also known as arctangent, which is defined for all real numbers.
3. The function cos⁡−1(x)\cos^{-1}(x) is the inverse cosine function, also known as arccosine, which is defined for −1≤x≤1-1 \leq x \leq 1.

STEP 2

1. Evaluate sin⁡−1(1.94)\sin^{-1}(1.94).
2. Evaluate tan⁡−1(2.27)\tan^{-1}(2.27).
3. Evaluate cos⁡−1(−0.62)\cos^{-1}(-0.62).

STEP 3

Evaluate sin⁡−1(1.94)\sin^{-1}(1.94):
The value 1.941.94 is outside the domain of the inverse sine function, which is [−1,1][-1, 1]. Therefore, sin⁡−1(1.94)\sin^{-1}(1.94) is undefined.

STEP 4

Evaluate tan⁡−1(2.27)\tan^{-1}(2.27):
The inverse tangent function is defined for all real numbers, so we can find tan⁡−1(2.27)\tan^{-1}(2.27) using a calculator or table of values. The approximate value is:
tan⁡−1(2.27)≈1.15\tan^{-1}(2.27) \approx 1.15 radians.

STEP 5

Evaluate cos⁡−1(−0.62)\cos^{-1}(-0.62):
The value −0.62-0.62 is within the domain of the inverse cosine function, which is [−1,1][-1, 1]. We can find cos⁡−1(−0.62)\cos^{-1}(-0.62) using a calculator or table of values. The approximate value is:
cos⁡−1(−0.62)≈2.24\cos^{-1}(-0.62) \approx 2.24 radians.
The solutions are:
sin⁡−1(1.94)=undefined\sin^{-1}(1.94) = \text{undefined}
tan⁡−1(2.27)≈1.15\tan^{-1}(2.27) \approx 1.15
cos⁡−1(−0.62)≈2.24\cos^{-1}(-0.62) \approx 2.24

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