Find all values of theta given that theta is between 0 degrees and 360 degrees. It is known that cos theta = -√3
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Timmy, Ryland, Cobra, Timmy -- you must be having an identity crisis tonight.
Please use the same name for all of your posts.
cosØ = -√3/2
1. find the reference angle
(using the positive result √3/2 , use inverse cos to get and angle of 30°
( Make yourself familiar with the ratio of sides of the 30-60-90 triangle)
2. Where is the angle Ø ?
Since the cosine is negative in II and III
in II : Ø = 180-30 or 150°
or
in III : Ø = 180+30 = 210°
oh so we only want reference angle. i thought values meant sin,cos,and tan. how do you know it only wanted reference angle?
didn't it say,
"Find the values of Ø" , not find the values of sinØ etc ?
Finding the reference angle is only the first step, from there you have to decide what to do with that reference angle depending on the position of Ø and the values of cosØ
To find all values of theta, given that cos(theta) = -√3/2, you can use the inverse cosine function, or arccos, to determine the angles for which the cosine function equals -√3/2.
The inverse cosine function, or arccos, is denoted as cos^(-1) or acos. For the given value of -√3/2, you can write it as acos(-√3/2).
To evaluate angles in the range between 0 and 360 degrees, you need to find the principal value of theta using the inverse cosine function.
Using a calculator in degree mode, evaluate acos(-√3/2) to find the principal value. In this case, the principal value of theta is 150 degrees.
Since cosine is a periodic function, you need to consider other angles that have the same cosine value as the principal value. To determine these angles, you can use the properties of the cosine function.
Since cosine is negative in the second and third quadrants, and it equals -√3/2 in the second quadrant, you can determine another angle with the same cosine value by subtracting the principal value from 180 degrees.
180 degrees - 150 degrees = 30 degrees
So, theta can also be 30 degrees.
Similarly, since cosine is negative in the third quadrant, and it equals -√3/2, you can determine another angle with the same cosine value by subtracting the principal value from 360 degrees.
360 degrees - 150 degrees = 210 degrees
So, theta can also be 210 degrees.
Hence, the values of theta between 0 degrees and 360 degrees, such that cos(theta) = -√3/2, are 150 degrees, 30 degrees, and 210 degrees.