Trigonometry ratio. Find the values of theater between 0° and 360°which satisfy each of the following . A.sin theater= 0.4396 B.cos theater=0.8427 C.tan theater= -2.4398 D.tan theater= 0.4926
A. sin θ = 0.4396
Using a calculator, we can find that the inverse sine of 0.4396 is approximately 25.98 degrees. However, sine is positive in both the first and second quadrants, so we also need to find the reference angle in the second quadrant which is 180 - 25.98 ≈ 154.02 degrees. Therefore, the solutions are θ ≈ 25.98 degrees and θ ≈ 154.02 degrees.
B. cos θ = 0.8427
Using a calculator, we can find that the inverse cosine of 0.8427 is approximately 31.94 degrees. However, cosine is positive in both the first and fourth quadrants, so we also need to find the reference angle in the fourth quadrant which is 360 - 31.94 ≈ 328.06 degrees. Therefore, the solutions are θ ≈ 31.94 degrees and θ ≈ 328.06 degrees.
C. tan θ = -2.4398
Using a calculator, we can find that the inverse tangent of -2.4398 is approximately -67.98 degrees. However, tangent is negative in the second and fourth quadrants, so we also need to add 180 degrees to get the solution in the second quadrant which is 180 - 67.98 ≈ 112.02 degrees. Also, we need to add 360 degrees to get the solution in the fourth quadrant which is 360 - 67.98 ≈ 292.02 degrees. Therefore, the solutions are θ ≈ 112.02 degrees and θ ≈ 292.02 degrees.
D. tan θ = 0.4926
Using a calculator, we can find that the inverse tangent of 0.4926 is approximately 25.87 degrees. However, tangent is positive in the first and third quadrants, so we also need to add 180 degrees to get the solution in the third quadrant which is 180 + 25.87 ≈ 205.87 degrees. Therefore, the solutions are θ ≈ 25.87 degrees and θ ≈ 205.87 degrees.
Let's solve each of the trigonometric equations step-by-step:
A. sin theater = 0.4396
To find the values of theater, we need to take the inverse sine (or arcsin) of 0.4396.
sin^(-1)(0.4396) ≈ 26.88°
Since sine is positive in Quadrant I and Quadrant II, we can find the theater values in both regions:
1) theater ≈ 26.88° (in Quadrant I)
2) theater ≈ 180° - 26.88° ≈ 153.12° (in Quadrant II)
Therefore, the values of theater that satisfy sin theater = 0.4396 are approximately 26.88° and 153.12°.
B. cos theater = 0.8427
To find the values of theater, we need to take the inverse cosine (or arccos) of 0.8427.
cos^(-1)(0.8427) ≈ 32.17°
Since cosine is positive in Quadrant I and Quadrant IV, we can find the theater values in both regions:
1) theater ≈ 32.17° (in Quadrant I)
2) theater ≈ 360° - 32.17° ≈ 327.83° (in Quadrant IV)
Therefore, the values of theater that satisfy cos theater = 0.8427 are approximately 32.17° and 327.83°.
C. tan theater = -2.4398
To find the values of theater, we need to take the inverse tangent (or arctan) of -2.4398.
tan^(-1)(-2.4398) ≈ -68.27°
Since tangent is negative in Quadrant II and Quadrant IV, we can find the theater values in both regions:
1) theater ≈ 180° - 68.27° ≈ 111.73° (in Quadrant II)
2) theater ≈ 360° + (-68.27°) ≈ 291.73° (in Quadrant IV)
Therefore, the values of theater that satisfy tan theater = -2.4398 are approximately -68.27°, 111.73°, and 291.73°.
D. tan theater = 0.4926
To find the values of theater, we need to take the inverse tangent (or arctan) of 0.4926.
tan^(-1)(0.4926) ≈ 25.17°
Since tangent is positive in Quadrant I and Quadrant III, we can find the theater values in both regions:
1) theater ≈ 25.17° (in Quadrant I)
2) theater ≈ 180° + 25.17° ≈ 205.17° (in Quadrant III)
Therefore, the values of theater that satisfy tan theater = 0.4926 are approximately 25.17° and 205.17°.
In summary:
A. sin theater = 0.4396:
- theater ≈ 26.88° and 153.12°
B. cos theater = 0.8427:
- theater ≈ 32.17° and 327.83°
C. tan theater = -2.4398:
- theater ≈ -68.27°, 111.73°, and 291.73°
D. tan theater = 0.4926:
- theater ≈ 25.17° and 205.17°