Theta = x
Question: Given angle x, where 0degrees <= x <= 360 degrees, solve for x to the nearest degree.
b) sin(x + 20degrees) = 0.2045
c) tan(90degrees - 2x) = 1.6443
I am confused as to where 148 degrees comes from in question b and how there are 4 possibilities in question c.
To solve these trigonometric equations, we will need to use the inverse trigonometric functions: arcsin, arccos, and arctan. These functions allow us to find the angle (in degrees) that corresponds to a given trigonometric ratio.
b) To solve sin(x + 20°) = 0.2045, we can rearrange the equation to isolate x:
x + 20° = arcsin(0.2045)
Now, to find x, we need to take the inverse sine of 0.2045. Using a calculator or a trigonometric table, we can find that arcsin(0.2045) ≈ 11.76°.
Since x + 20° = 11.76°, we can solve for x:
x = 11.76° - 20° ≈ -8.24°
However, the given range for x is 0° ≤ x ≤ 360°. To find a value within this range that satisfies the equation, we need to add or subtract multiples of 360°. So:
x = -8.24° + 360° ≈ 351.76°
Rounding to the nearest degree, x ≈ 352°. So, there was an error in stating that x = 148°.
c) To solve tan(90° - 2x) = 1.6443, let's rearrange the equation first:
90° - 2x = arctan(1.6443)
Now, we can take the inverse tangent of 1.6443 to find the angle:
arctan(1.6443) ≈ 58.75°
Next, solve for x:
90° - 2x = 58.75°
Subtract 58.75° from 90° and divide by 2:
-2x = 31.25°
x = 31.25° / -2 ≈ -15.63°
Again, since the given range for x is 0° ≤ x ≤ 360°, we need to find multiple solutions within this range.
To do this, let's consider the periodic nature of the tangent function. Tan(x) repeats itself every 180°:
tan(x) = tan(x + 180°)
Using this property, we can find the additional possible solutions:
x = -15.63° + 180° ≈ 164.37°
x = 164.37° + 180° ≈ 344.37°
So, there are actually four possible solutions: -15.63°, 164.37°, 344.37° (within the range 0° to 360°).
I hope this clarifies where the values of 148° and the four possibilities in question c come from. Let me know if you have any further questions!
To solve the equations, we will need to use trigonometric identities and solve for x. Let's start with question b.
b) sin(x + 20 degrees) = 0.2045
To find the value of x, we first need to isolate x on one side of the equation. To do this, we can use the inverse sine function (also known as arcsin or sin^-1) on both sides of the equation:
sin^(-1)(sin(x + 20 degrees)) = sin^(-1)(0.2045)
The inverse sine function cancels out the sine function, leaving us with:
x + 20 degrees = sin^(-1)(0.2045)
Now we can substitute the value for sin^(-1)(0.2045) from a calculator:
x + 20 degrees ≈ 11.93
Subtracting 20 degrees from both sides:
x ≈ 11.93 - 20
x ≈ -8.07
Since we are given that 0 degrees ≤ x ≤ 360 degrees, we can add 360 degrees to -8.07 to find another possible angle:
x ≈ -8.07 + 360
x ≈ 351.93
Therefore, the possible values of x to the nearest degree are approximately -8 degrees and 352 degrees.
Regarding question c:
c) tan(90 degrees - 2x) = 1.6443
To solve this equation, we first need to isolate the tangent function by taking the inverse tangent (arctan) of both sides:
arctan(tan(90 degrees - 2x)) = arctan(1.6443)
Since the tangent function has a periodicity of 180 degrees, we can simplify the left side:
90 degrees - 2x + 180 degrees = arctan(1.6443)
270 degrees - 2x = arctan(1.6443)
Next, we can substitute arctan(1.6443) with its approximate value:
270 degrees - 2x ≈ 59.999
Solving for x:
-2x ≈ 59.999 - 270
-2x ≈ -210.001
Dividing both sides by -2:
x ≈ -210.001 / -2
x ≈ 105.0005
Since we are given that 0 degrees ≤ x ≤ 360 degrees, let's find all possible values of x:
x ≈ 105.0005 + 180
x ≈ 285.0005
x ≈ 105.0005 + 360
x ≈ 465.0005
Therefore, to the nearest degree, the possible values of x are approximately 105 degrees, 285 degrees, 465 degrees, and their corresponding supplementary angles.