1) Find two values of Q that satisfy the equation. Give your answer is degrees and radians. Do not use a calculator.
a) sec Q = 2
-pi/3 and 60 degrees
Is this correct?
b) sec Q = -2
This one I do not know how to find.
To solve the equation sec(Q) = 2, we need to find the values of Q in both degrees and radians. The secant function is the reciprocal of cosine, so we can rewrite the equation as cos(Q) = 1/2.
To find the values of Q, we can use the unit circle, which represents the values of the trigonometric functions for angles from 0 to 360 degrees (0 to 2π radians).
a) For sec(Q) = 2 (or cos(Q) = 1/2), we know that the cosine function equals 1/2 for two particular angles: 60 degrees and 300 degrees (or π/3 and 5π/3 radians). These angles are based on the reference angle of 60 degrees (π/3 radians) in the first quadrant, and its cosine value is positive in both the first and fourth quadrants.
Therefore, the two values of Q that satisfy sec(Q) = 2 are Q = 60 degrees and Q = 300 degrees (or Q = π/3 radians and Q = 5π/3 radians).
In your given answer, you correctly identified one of the values as 60 degrees, but the other value should be 300 degrees (not -60 degrees).
b) Moving on to sec(Q) = -2 (or cos(Q) = -1/2), we again need to find the angles where the cosine function is -1/2. Since the cosine function is negative in the second and third quadrants, we need to look for the angles in those quadrants.
For the cosine function to be -1/2, we need to find angles that have the same reference angle as 60 degrees (π/3 radians), but with a negative cosine value.
In the unit circle, we find that the cosine function equals -1/2 for two specific angles: 120 degrees and 240 degrees (or 2π/3 and 4π/3 radians). These angles are based on the 60 degrees (π/3 radians) reference angle in the second and third quadrants.
Therefore, the two values of Q that satisfy sec(Q) = -2 are Q = 120 degrees and Q = 240 degrees (or Q = 2π/3 radians and Q = 4π/3 radians).
To summarize:
- For sec(Q) = 2, Q can be 60 degrees (π/3 radians) and 300 degrees (5π/3 radians).
- For sec(Q) = -2, Q can be 120 degrees (2π/3 radians) and 240 degrees (4π/3 radians).