View
23
Download
2
Category
Preview:
DESCRIPTION
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? - PowerPoint PPT Presentation
Citation preview
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.
2
1sin
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.
2
1sin
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
2
1sin
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.
02cos x
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.
02cos x
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
02cos x
Use it
Combined general solution: θ = π/2 + πn
Final step, remove the substitution and solve for x.
02cos 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.
02cos x
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?
xxx sintansin
Use it
Incorporate factoring to solve for sin θ and tan θ.
Now, solve for the angles.
1tan
0sin
0)1(tansin
0sintansin
sintansin
x
x
xx
xxx
xxx
Use it
Incorporate factoring to solve for sin θ and tan θ.
tan θ = 1 sin θ = 0
One rotation θ = π/4, 5π/4 θ = 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.
1tan
0sin
x
x
Use it
Remember the trig identities. Factor again.
01cossin2 2 tt
Use it
Remember the trig identities. Factor again.
1cos2
1cos
0)1)(cos1cos2(
01coscos2
01coscos2
01coscos22
01cos)cos1(2
01cossin2
2
2
2
2
2
t
t
tt
tt
tt
tt
tt
tt
Use it
Now solve for t.
1cos2
1cos
t
t
Use it
Now solve for t.
cos t = ½ cos t = -1
One rotation t = π/3, 5π/3 t = π
General sol. t = π/3 ± 2πn t = π ± 2πn
t = 5π/3 ±2πn
(Combined: t = ±π/3 ± 2πn)
1cos2
1cos
t
t
Recommended