Trigonometric Equations Solving for the angle (The first of two note days and a work day) (6.2)(1)

POD Solve for angles in one rotation, then for a general solution.

Trig Equations What makes it an equation? What makes it a trig equation? We’re going to use a lot of inverse trig functions today. Note: Unless otherwise specified, operate in radians.

Trig Equations Steps to solve them: 1. Solve the equation for sinθ, cosθ, or tanθ. 2. Find values of θ that satisfy the equation in one rotation. 3. Consider all possible values of θ for a general solution. 4. If needed, undo any substitutions and solve for any variables. We’ve done some of this already when we used inverse trig functions, say, in the POD.

Use it A riff on the POD. Step 1 is done. Step 2 Solve for 0 ≤ θ ≤ 2π. Step 3 Solve for all θ.

Use it A riff on the POD. Step 2 Solve for 0 ≤ θ ≤ 2π. θ = 7π/6 and 11π/6 Step 3 Solve for all θ. θ = 7π/6 ± 2πn and 11π/6 ± 2πn

Use it A step beyond– solve for the angle, then for the variable (step 4 in the method). In this case, find the general solution and then give all values of x in the interval 0 ≤ x ≤ 2π.

Use it A step beyond– solve for the angle, then for the variable (step 4 in the method). First step is done. Second step, solve for 1 rotation, then a general solution.

Use it Second step, solve for 1 rotation to build a general solution. In one rotation: θ = π/2 and θ = 3π/2. (Notice how I substituted θ for 2x; it’s easier to work with.) Third step, general solution: θ = π/2 ± 2πn and θ = 3π/2 ± 2πn Combined general solution: π/2 + πn

Use it Combined general solution: θ = π/2 + πn Final step, remove the substitution and solve for x.

Use it Combined general solution: θ = π/2 + πn From the general solution θ = 2x = π/2 ± πn x = π/4 ± πn/2 So, in the interval 0 ≤ x ≤ 2π, x = π/4, 3π/4, 5π/4, 7π/4.

Use it Combined general solution: θ = π/2 + πn x = π/4 ± πn/2 Check: Compare the graph of y = cos x to y = cos 2x. What changes? What are the x-intercepts? How does this graph relate to our solution?

Use it Incorporate factoring to solve for sin θ and tan θ. What should you NOT do?

Use it Incorporate factoring to solve for sin θ and tan θ. Now, solve for the angles.

Use it Incorporate factoring to solve for sin θ and tan θ. tan θ = 1sin θ = 0 One rotationθ = π/4, 5π/4θ = 0, π General sol. θ = π/4 ± πn θ = ±πn This means that any angle in either category will make the equation true. Test it with θ = π and π/4.

Use it Remember the trig identities. Factor again.

Use it Remember the trig identities. Factor again.

Use it Now solve for t.

Use it Now solve for t. cos t = ½cos t = -1 One rotationt = π/3, 5π/3t = π General sol.t = π/3 ± 2πnt = π ± 2πn t = 5π/3 ±2πn (Combined: t = ±π/3 ± 2πn)