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.
b) sin(x + 20degrees) = 0.2045
Using your calculator you should get:
x+20° = 11.8 in quadI OR x+20 = 168.2 in quadII
x = -8.2° , but our domain was 0 ≤ x ≤ 360° , so we reject that answer
or
x = 168.2-20 = 148.2° , the given answer
c) tan(90degrees - 2x) = 1.6443
90° -2x = 58.7° in quad I or 90-2x = 238.7° in quad III
-2x = -31.3 or -2x = 148.7
x = 15.65 or x = - -74.35° <----- not admissable
now the period of tanØ = 90°
so adding multiples of 90 to any of our answers will yield a new one, so ...
x = - 74.35 + 90 = 15.65°
x = 15.65°
x = 15.65 + 90 = 105.65°
x = 105.65+90 = 195.65°
x = 195.65+90 = 285.65 <------ there are your 4 answers
15.65, 105.65, 195.65 , 285.65
Well, looks like we have a couple of interesting questions here. Let's break it down!
For question b, we're looking to solve sin(x + 20 degrees) = 0.2045. Now, to find the value of x, we need to look for the angles where sin(x + 20 degrees) equals 0.2045. One solution is indeed x = 148 degrees. But remember, we have a sine function, and it periodically repeats itself. So if you continue around the unit circle, you'll find three more solutions that differ by multiples of 360 degrees. So, x could be 148 degrees, or 508 degrees, or 868 degrees, or even -212 degrees, if we're feeling a bit negative!
Now, question c is a bit trickier. We're solving tan(90 degrees - 2x) = 1.6443. Here, we know that tan(x) has periodicity of pi (180 degrees), with a vertical asymptote every 180 degrees. So if we subtract 90 degrees from 2x, the values we get for x will be halfway between these vertical asymptotes. Therefore, there are four possibilities within the interval 0 degrees to 360 degrees. However, I must warn you that I wouldn't mess with a clown trying to solve this equation – it sounds like a circus!
I hope that clarifies things a bit, even if it may seem a bit confusing. Don't hesitate to ask if you have any more questions, I'm here to entertain!
b) To solve sin(x + 20 degrees) = 0.2045, we need to find the value of x within the given range where the sine function equals 0.2045.
Step 1: Start by subtracting 20 degrees from both sides of the equation to isolate x:
sin(x + 20 degrees) - sin(20 degrees) = 0.2045 - sin(20 degrees)
Step 2: Use the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) to simplify the left side of the equation:
sin(x)cos(20 degrees) + cos(x)sin(20 degrees) - sin(20 degrees) = 0.2045 - sin(20 degrees)
Step 3: Rearrange the equation:
sin(x)cos(20 degrees) + cos(x)sin(20 degrees) = 0.2045 - sin(20 degrees)
Step 4: Use the sine addition formula to rewrite the left side of the equation:
sin(x + 20 degrees) = 0.2045 - sin(20 degrees)
Step 5: Now, using a scientific calculator or a trigonometric table, find the angle whose sine value is 0.2045. In this case, it is approximately 11.78 degrees.
Step 6: Substitute the value of sin(20 degrees) and solve for x:
sin(x + 20 degrees) = 0.2045 - sin(20 degrees)
sin(x + 20 degrees) = 0.2045 - 0.3420
sin(x + 20 degrees) = -0.1375
Step 7: From the unit circle or trigonometric table, you can find the two angles x + 20 degrees that have a sine value of -0.1375 (approximately). These two angles are approximately -148 degrees and 32 degrees.
Therefore, the solutions to sin(x + 20 degrees) = 0.2045 for x, rounded to the nearest degree, are -148 degrees and 32 degrees.
c) To solve tan(90 degrees - 2x) = 1.6443, we need to find the values of x within the given range where the tangent function is equal to 1.6443.
Step 1: Start by subtracting 1.6443 from both sides of the equation to isolate the tangent function:
tan(90 degrees - 2x) - 1.6443 = 0
Step 2: Convert the equation to a form without trigonometric functions by using the identity tan(A) = sin(A)/cos(A):
sin(90 degrees - 2x) / cos(90 degrees - 2x) - 1.6443 = 0
Step 3: Simplify further:
cos(2x - 90 degrees) / sin(2x - 90 degrees) - 1.6443 = 0
Step 4: Multiply both sides of the equation by sin(2x - 90 degrees) to eliminate the denominator:
cos(2x - 90 degrees) - 1.6443sin(2x - 90 degrees) = 0
Step 5: Rearrange the equation:
1.6443sin(2x - 90 degrees) = cos(2x - 90 degrees)
Step 6: Divide both sides of the equation by cos(2x - 90 degrees) to isolate sin(2x - 90 degrees):
1.6443sin(2x - 90 degrees) / cos(2x - 90 degrees) = 1
Step 7: Recognize that the left side of the equation is the tangent function, so simplify further:
tan(2x - 90 degrees) = 1
Step 8: Solve for 2x - 90 degrees. The tangent function equals 1 when the angle is 45 degrees.
Step 9: Set 2x - 90 degrees equal to 45 degrees and solve for x:
2x - 90 degrees = 45 degrees
Step 10: Add 90 degrees to both sides of the equation:
2x = 135 degrees
Step 11: Divide both sides of the equation by 2:
x = 67.5 degrees
Since the given range is 0 degrees to 360 degrees, there are four solutions within this range:
x = 67.5 degrees
x = 67.5 + 180 degrees = 247.5 degrees
x = -67.5 degrees
x = -67.5 + 180 degrees = 112.5 degrees
Therefore, there are four possible values for x in the equation tan(90 degrees - 2x) = 1.6443, rounded to the nearest degree: 68 degrees, 112 degrees, 248 degrees, and 292 degrees.
To solve these trigonometric equations, we need to use inverse trigonometric functions (also known as arc functions) to find the value of the angles. We'll go step by step for both questions.
b) sin(x + 20 degrees) = 0.2045
Step 1: Use the inverse sine function (sin^(-1)) to isolate the angle.
sin^(-1)(sin(x + 20 degrees)) = sin^(-1)(0.2045)
x + 20 degrees = sin^(-1)(0.2045)
Step 2: Evaluate the inverse sine function on the right side using a calculator.
x + 20 degrees ≈ 11.79 degrees
Step 3: Solve for x by subtracting 20 degrees from both sides.
x ≈ 11.79 degrees - 20 degrees
x ≈ -8.21 degrees
Since the question specifies that x lies between 0 and 360 degrees, we need to find the equivalent positive angle within that range.
x ≈ -8.21 degrees + 360 degrees
x ≈ 351.79 degrees
Hence, to the nearest degree, x is approximately 352 degrees.
c) tan(90 degrees - 2x) = 1.6443
Step 1: Use the inverse tangent function (tan^(-1)) to isolate the angle.
tan^(-1)(tan(90 degrees - 2x)) = tan^(-1)(1.6443)
90 degrees - 2x = tan^(-1)(1.6443)
Step 2: Evaluate the inverse tangent function on the right side using a calculator.
90 degrees - 2x ≈ 57.89 degrees
Step 3: Solve for x by subtracting 90 degrees and dividing by -2.
-2x ≈ 57.89 degrees - 90 degrees
-2x ≈ -32.11 degrees
x ≈ (-32.11 degrees) / (-2)
x ≈ 16.06 degrees
Since the question asks for all possible solutions, we need to account for the periodicity of the tangent function, which repeats every 180 degrees. So we can add 180 to any value of x to find additional solutions.
All possible solutions to the equation are:
x ≈ 16.06 degrees + 180 degrees (196.06 degrees)
x ≈ 16.06 degrees + 2 * 180 degrees (376.06 degrees)
x ≈ 196.06 degrees - 180 degrees (16.06 degrees)
Hence, there are four possibilities: 16, 196, 376, and 556 degrees.