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The same ratio of sides happens at 150°, 390°, 510°, etc.Not only that, you can rotate backward (clockwise), and get negative angles. Because of this, most trig problems include a domain limitation (upper and lower limits for possible inputs).
The following identities are essential to all your work with trig functions. Quotient Identities: tan(x) = sin(x)/cos(x) cot(x) = cos(x)/sin(x) Reciprocal Identities: csc(x) = 1/sin(x) sec(x) = 1/cos(x) cot(x) = 1/tan(x) sin(x) = 1/csc(x) cos(x) = 1/sec(x) tan(x) = 1/cot(x) Pythagorean Identities: sin The following seven step process will work every time.
It is rather tedious, and can take more time than necessary.
I use Scientific Notebook or similar math software to graph the functions for me.
You can use this Online Graphing Calculator to solve the following equations (or check your solutions) .
For example, domains might be limited to 0° to 180°, 0° to 360°, -180° to 180°, etc.
As a side note, remember that some trig problems work in radians instead of degrees. Radians use the π properties of a circle, and instead of expressing angles as part of a 360° circle, they express angles as a part of a 2π circle.Some teachers will ask you to prove the identity directly (from one side to the other in a straight line). Bourne Trigonometric equations can be solved using the algebraic methods and trigonometric identities and values discussed in earlier sections.As you gain more practice, you can skip or combine these steps when you recognize other identities.STEP 1: Convert all sec, csc, cot, and tan to sin and cos.The following is a simple example: 30° is just one of the solutions.An angle is a measure of rotation, and if you keep rotating you'll get more angles with the same sine.STEP 3: Check for angle multiples and remove them using the appropriate formulas.STEP 4: Expand any equations you can, combine like terms, and simplify the equations.So the only solution for this part is `x=(2pi)/3.` Also, `cos x=-1` gives `x = pi`. So the solutions for the equation are `x=(2pi)/3or pi.` A check of the graph of `y=cos x/2-1-cos x` confirms these results: Note 1: "Analytically" means use the methods and formulas from previous sections. Note 2: However, I always use a graph to check my analytical work.