Given angle theta is bigger than 0 and smaller than 360, determine two possible values of theta where each ration would be true.
a) cos theta= 0.6951
b)sin theta= -0.7571
Here is basically the simplest way to do these type of questions.
I will assume you are familiar with the CAST rule and have a calculator.
1. Use the absolute value of the given constant and use the inverse of the given trig ratio.
in this case, enter
2nd cos .6951 and press the = sign.
you should get 45.9648...
That is your "angle in standard position" , namely the first quadrant angle.
2. The cosine is positive in I and IV
so Ø = 45.965° or 360-45.965= 314.035°
b)
enter 2nd sin .7571 and press = to get
49.21°
BUT our sine was negative , so Ø had to be in III or IV
in III Ø = 180+49.21 = 229.21
in IV Ø = 360-49.21 = 310.79°
check these answers with your calculator, they work.
a) To find the two possible values of theta for the given cosine ratio, we can use the inverse cosine function (cos^(-1)).
cos(theta) = 0.6951
Step 1: Take the inverse cosine of both sides:
cos^(-1)(cos(theta)) = cos^(-1)(0.6951)
Step 2: Simplify:
theta = cos^(-1)(0.6951)
Using a calculator or reference table, we find that the approximate value of cos^(-1)(0.6951) is 45.14 degrees.
Therefore, one possible value of theta is 45.14 degrees.
To find the second possible value of theta, we can use the fact that the cosine function has a periodicity of 360 degrees.
Step 3: Subtract the first value of theta from 360:
360 - 45.14 = 314.86 degrees
Therefore, the second possible value of theta is 314.86 degrees.
b) To find the two possible values of theta for the given sine ratio, we can use the inverse sine function (sin^(-1)).
sin(theta) = -0.7571
Step 1: Take the inverse sine of both sides:
sin^(-1)(sin(theta)) = sin^(-1)(-0.7571)
Step 2: Simplify:
theta = sin^(-1)(-0.7571)
Using a calculator or reference table, we find that the approximate value of sin^(-1)(-0.7571) is approximately -48.71 degrees.
Therefore, one possible value of theta is -48.71 degrees.
To find the second possible value of theta, we can use the fact that the sine function has a periodicity of 360 degrees.
Step 3: Add 360 to the first value of theta:
-48.71 + 360 = 311.29 degrees
Therefore, the second possible value of theta is 311.29 degrees.
To find the possible values of theta for each equation, we need to use inverse trigonometric functions. Specifically, for the equation "cos theta = 0.6951," we will use the inverse cosine function (also known as arccosine or cos^-1), and for the equation "sin theta = -0.7571," we will use the inverse sine function (also known as arcsine or sin^-1).
a) cos theta = 0.6951:
To find the possible values of theta, we need to use the inverse cosine function:
theta = cos^-1(0.6951)
Now, to solve this using a calculator:
1. Make sure your calculator is in radian mode.
2. Press the inverse cosine button (typically labeled as "cos^-1" or "arccos").
3. Enter the value 0.6951.
4. Press the equal (=) button.
The calculator will then give you the result, which corresponds to one possible value of theta. However, keep in mind that this will only give you one solution in the range [0, π]. Since there are two possible values for theta, we can find the other one by utilizing the symmetry of the cosine function.
b) sin theta = -0.7571:
To find the possible values of theta, we need to use the inverse sine function:
theta = sin^-1(-0.7571)
Similar to the previous equation, we need to use a calculator to find the value of theta:
1. Ensure your calculator is in radian mode.
2. Press the inverse sine button (typically labeled as "sin^-1" or "arcsin").
3. Enter the value -0.7571.
4. Press the equal (=) button.
Again, the calculator will provide you with one solution within the range [-π/2, π/2]. Since there are two possible values for theta, we can find the other one by utilizing the symmetry of the sine function.
It is worth noting that these equations may have multiple solutions, as trigonometric functions have a periodic nature. However, since you specified that theta is bigger than 0 and smaller than 360, we only consider the values within that range.