Definition: To solve trigonometric equations, we’ll need to combine our knowledge of the unit circle, trigonometric functions, and trigonometric identities with algebraic methods of solving.
Example 1: Find all solutions of cos(2x) = 0 in the interval [0, 2π).
Step 1: Recall where the cosine function has a value of 0. We know this occurs on the y-axis of the unit circle at π/2, 3π/2, and for any value π/2 + nπ (where n is an integer).
Looking at the first value, π/2, we can write the following equation and solve for x:
Step 2: Repeat this step to find all solutions in the given interval, [0, 2π), keeping in mind the general solution π/2 + nπ :
This gives us the following solutions:
Example 2: Find all solutions:
Step 1: We will need to solve the sine function as if it were a quadratic, so can start by factoring and applying the zero product principle:
Step 2: Solve both equations for x:
sin(x) = ±√3/2 when x = π/3, 2π/3, 4π/3, 5π/3, and all values given by the general solutions π/3 + nπ and 2π/3 + nπ (where n is an integer).
sin(x) = ±1/2 when x = π/6, 5π/6, 7π/6, 11π/6, and all values given by the general solutions π/6 + nπ and 5π/6 + nπ, (where n is an integer).
We add multiples of π because every 2π, it’s the same angle and every odd π gives the additive inverse of the value.
Step 3: Express all of the possible solutions found in Step 2 as the final solution:
, (where n is an integer).
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