1) Find two values of o that satisfy the equation. Give your answers in degrees and radians. Do not use a calculator.
(a) sec o = 2
(b) sec 0 = -2
sec(x) = 1/cos(x). So all you need to do is find where cos(x) = 1/2 or 1/(-2).
To solve these equations without using a calculator, we will need to use the unit circle and the properties of trigonometric functions.
(a) For the equation sec o = 2:
Recall that secant is the reciprocal of cosine: sec o = 1 / cos o.
We need to find an angle o whose cosine is equal to 1/2. Looking at the unit circle, we can see that this happens when o is an angle in the first or second quadrants where the x-coordinate is positive.
In the unit circle, the cosine values for angles in the first quadrant (0° to 90° or 0 to π/2 radians) are positive. Therefore, we can set up the equation cos o = 1/2.
To find the two values of o that satisfy this equation, we can look at the special triangle with sides 1, 2, and √3. This triangle corresponds to the angle 60° (or π/3 radians).
Since cosine is equal to the adjacent side divided by the hypotenuse, we have cos 60° = 1/2. Therefore, o = 60° (or π/3 radians) is one solution.
To find the second solution, we notice that in the second quadrant (90° to 180° or π/2 to π radians), the cosine values are also positive. Hence, we can use the symmetry property of cosine to find the second solution.
Since the cosine function is an even function, we know that cos(-θ) = cos(θ), where θ represents any angle. Thus, if θ is a solution, then -θ is also a solution.
For our equation, we have cos o = 1/2. If o = 60° (or π/3 radians) is a solution, then -o = -60° (or -π/3 radians) must also be a solution.
Therefore, the two values of o that satisfy sec o = 2 are o = 60° and o = -60° (or π/3 radians and -π/3 radians).
(b) For the equation sec 0 = -2:
Using the same reasoning as above, we know that secant is equal to the reciprocal of cosine.
Since the secant is -2, this implies that the cosine is -1/2. Looking at the unit circle, we can see this happens when the angle is in the second or third quadrants where the x-coordinate is negative.
In the unit circle, the cosine values for angles in the second quadrant (90° to 180° or π/2 to π radians) are negative.
To find an angle whose cosine is -1/2, we can use the same special triangle as before (with sides 1, 2, and √3) to find the angle.
Since cosine is equal to the adjacent side divided by the hypotenuse, we have cos 120° = -1/2. Therefore, 0 = 120° (or -π/3 radians) is one solution.
By using the symmetry property of cosine, we can find the second solution. In the third quadrant (180° to 270° or π to 3π/2 radians), the cosine values are negative. Therefore, we can use the symmetry to find the second solution.
If θ is a solution, then -θ is also a solution. So, if θ = 120° (or -π/3 radians) is a solution, then -θ = -120° (or 2π/3 radians) must also be a solution.
Therefore, the two values of o that satisfy sec 0 = -2 are o = 120° and o = -120° (or -π/3 radians and 2π/3 radians).