*You may wish to go back and have a look at Trigonometric Functions of Any Angle, where we see the background to the following solutions. Where the graph cuts the x-axis, that's where you'll find your solutions (the x-values that "work").*Graphs also help you to understand why sometimes there is one answer, and sometimes many answers.I use Scientific Notebook or similar math software to graph the functions for me.

*You may wish to go back and have a look at Trigonometric Functions of Any Angle, where we see the background to the following solutions. Where the graph cuts the x-axis, that's where you'll find your solutions (the x-values that "work").*

We proceed as follows: We solve We can see from the graph that our 4 values are reasonable, since these are the only 4 values that satisfy `sin 2θ = 0.8`.

Note: We can always check our solutions with calculator, but it is easy to "miss out" on some of the required values if we only use calculator.

If you've been working in trigonometry, you've probably seen sines, cosines, tangents, and angle relationships until it's nearly driven you nuts.

Before you lock yourself in a psych ward somewhere, let's take a look at some of them, and see if we can't make them just a bit easier.

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.

Solve the equation If the problem involved θ only, we would expect 2 solutions; one in the first quadrant and one in the second quadrant.

But here our problem involves `2θ`, so we have to double the domain (θ values) to account for all possible solutions.

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.

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