I'm confused on how to determine what the end result angles are for a trigonometric identity (I get how to find the value of sin, cos, etc., and I know how to find the reference angle. I just don't know how to determine further angles from there).
For example, 3tan^2x-1 equals the square root of 3 over three. I know 30 degrees is my reference angle. My book says the other answer is 150. But where does this come from? As a general rule, do you find the second angle by subtracting the reference angle from 180 (or 360 if it's that sort of problem)? But this approach doesn't work for the negative sq. root of 2/2=sinx. 360-225 doesn't equal the second angle, 315. So is the way to find the subsequent angles by adding the reference angle (in this case 30) to the degree value of the nearest axis (90, 180, etc)?? Yet this doesn't work for a reference angle of the sq. root +/- 1/2=sinx. One of the answers is 330-270+30 doesn't equal this. Please help clarify this!
When solving trigonometric identities, finding the reference angle is indeed the first step, so it's great that you understand that part. To find the other angles that satisfy the identity, you need to consider a few things.
1. Determine the pattern: For trigonometric functions, the values repeat in a periodic pattern. For example, sin(x) repeats every 360 degrees (or 2π radians), and cos(x) also repeats every 360 degrees. For tangent, the pattern repeats every 180 degrees (or π radians).
2. Consider the relationship with the reference angle: The trigonometric functions have periodicity, so you need to find the relationship between the reference angle and other angles that satisfy the equation. This depends on the signs of the trigonometric functions in the given equation.
3. Apply the pattern: Once you find the relationship between the reference angle and other angles, you can apply the pattern to find the corresponding angles.
Let's apply this process to the examples you provided:
Example 1: 3tan^2x - 1 = sqrt(3)/3
In this case, the tangent has a positive value (sqrt(3)/3) in the first quadrant (from 0 to 90 degrees or 0 to π/2 radians), which means the reference angle is 30 degrees or π/6 radians. Since the pattern for tangent repeats every 180 degrees, you can add 180 degrees to the reference angle to find the other angle. Thus, the other angle is 30 + 180 = 210 degrees or π/6 + π = 7π/6 radians.
Example 2: sin(x) = -sqrt(2)/2
In this case, the sine function has a negative value (-sqrt(2)/2) in the third and fourth quadrants (180 to 270 degrees or π to 3π/2 radians, and 270 to 360 degrees or 3π/2 to 2π radians, respectively). The reference angle is 45 degrees or π/4 radians. To find the angles in the third and fourth quadrants, you need to consider the relationship with the reference angle. In the third quadrant, you add 180 degrees to the reference angle (45 + 180 = 225 degrees or 5π/4 radians). In the fourth quadrant, you subtract the reference angle from 360 degrees (360 - 45 = 315 degrees or 7π/4 radians).
So, to summarize, you find subsequent angles by applying the pattern for the specific trigonometric function and considering the relationship with the reference angle based on the signs of the trigonometric functions in the given equation. I hope this clarifies the process for determining the other angles in trigonometric identities!