Download 6.7 Day 1 – Solving Trig. Equations

Unit 9 Day 1 – Solving
Trig. Equations
We will solve simple trig equations
giving both general solutions and
solutions over an interval
and
I will practice solving these problems.
March 13, 2017
Trigonometric Equations
• Trigonometric identity - trig equation that holds
true for any angle
• Trigonometric equation - involves one or more
trigonometric functions and is only true for
certain angles
• To solve a trigonometric equation: use trig.
identities and algebraic techniques to reduce the
given equation to an equivalent but more
manageable expression.
Examples – Solve the equation
2 cos x  1  0
2cosx = -1
cosx = -½
2π 4π
x
,
3 3
where on the unit circle is
cosx = - ½?
Examples – Solve the equation
3 sec x  2
2
sec x  
3
3
cos x  
2
5π 7π
x
,
6 6
Examples – Solve the equation
3 cot x   3
3
cot x  
3
2π 5π
x
,
3 3
Examples – Solve the equation
2cos x  1  0
2
2cos x  1
2
1
cos x 
2
2
1
cos x  
2
2
cos x  
2
 3 5 7
x , , ,
4 4 4 4
Solve the equation
2 cos x  1  0
2 cos x  1
1
cos x 
2
2
cos x 
2
 7
x ,
4 4
Are these
the only
answers??
Solving Trig Equations
General rule: Because trig functions are periodic, if
there is one solution, then there are an infinite
number of solutions.
There are 2 ways to give a solution to a trig equation:
• Solutions in an interval: include all the solutions
that fall in an interval (usually 0 ≤ θ < 2π). This is
what we have been doing thus far.
• General Solution:
θ +2kπ (for sin, cos, sec, csc)
θ + kπ (for tan, cot)
where θ is one solution to the equation, and k is an
integer
Solve the equation. Give a general
formula for all solutions.
2sinθ + 3 = 2
2sinθ = -1
sinθ = -½
general solution:
7π
θ=
+ 2kπ
6
11π
θ=
+ 2kπ
6
7  11

,
6 6