How do we solve linear trigonometric equations?

Aim: How do we solve linear trigonometric equations? 

Solving trigonometric equations are easy as long as you know how to do them.  

There are three steps to solving linear equations:
1)  Get the trig function alone.
2)  Use inverse to solve for angle or use the helpful Unit Circle!
3)  Check for other solutions.

Example: 

Solve for all values between 0 to 2:
2 sin θ-√3 = 0
1) To get the trig function alone, add √3 to both sides of the equation.
                             2 sinθ -√3 = 0
                                       +√3   +√3
                             2 sinθ = √3
2) Divide 2 from both sides to get sinθ completely alone.
                              
                                 2 sinθ = √3
                                    2          2
                                 sinθ =  √3
                                              2
3) Now, if you look at the unit circle, you will see that the only places where sin (y-coordinate) is √3 are at π and 2π or in degrees 60° 
                                               2            3         3
and 120°. These are the solutions. You can also achieve these solutions by first using the calculator to solve for inverse sine of √3, 
                                                                                                       2
which will give you 60° and you can subtract it from 180 to get 120° since sine is only positive in the first two quadrants or 180°
Example #2:

tan θ -­ 2 cos θ tan θ = 0

1) Factor to solve for both tanθ and cosθ. Factor out tanθ since it is 

common. 

                                  tan θ -­ 2 cos θ tan θ = 0
                                  
                                   tanθ (1- 2 cosθ) = 0

2) Set each equal to 0 and solve.


                    tanθ = 0               and              1-2 cosθ = 0

                                                                     -1           -1


                                                                    -2 cosθ = -1

                                                                     -2            -2
                                                                      
                                                                      cosθ = 1/2




3) If you look at the unit circle, you will see that tangent (sin/cos) 

equals to 0 at 0° and 180° or 0 and π in radians. The cosine(x-

coordinate) are 60° and 300° or π/3 and 5π/3. These are the 

solutions. 

How do we graph the other trig functions?

Aim: How do we graph the other trig functions ?

To graph secant and cosecant, we fist graph sine and cosine to make our lives easier. After graphing cosine or sine, we can use the characteristics of secant and secant graphs to graph them.

1) Graph y=sinx
(The graph starts out at zero, goes up to positive 1 and than comes back to zero, later it goes down to negative 1 and again comes back to zero and continues in this pattern).  

2) Since cosecant is reciprocal of sine, every time sin of x equals to zero, you have an asymptote. Draw the asymtotes. 

y=cscx
 = 1/sinx
So, if 1/0, it is undefined. (Cannot have zero in the denominator).


- The asymptote occurs at pi and repeats every pi units. (That is where sine is zero).

3) The graph of csc comes on where the graph of sine does not equal to zero and stays in between the asymptotes. 

 -Remember it does not touch the asymptotes. It does not have an amplitude because the graph continues vertically and has a period of 2pi. 

The images keep on rotating for some weird reason. 

The same steps apply for the graph of secant, except you start out with graphing the cosine of x. (The graph of cosine starts from one)

Why the name Pythagorean Identity is appropriate?

Why the name Pythagorean Identity is appropriate? 

The Pythagorean Identities are trigonometric functions that are made using the Pythagorean Theorem.   The name Pythagorean Theorem is appropriate because the function is based on the Pythagorean Theorem.

The Pythagorean Theorem is the equation a^2+b^2=c^2, where a and b are the legs of a right triangle and c is the hypotenuse. 

Similarly, in a unit circle, there are right triangles. The radius of a unit circle is always 1 and it is the hypotenuse of the right triangle created by an angle. The legs of the triangle are the  length of the y and x axis. The length of the y axis is called sine and the length of the x axis is called cosine.

Now, the Pythagorean Theorem can be used to make an equation for the right triangle in the unit circle. 

cos^2θ+sin^2θ=1^2

or 

cos^2θ+sin^2θ=1

This equation if called Pythagorean Identity number 1.When this 

equation is solved in terms of one of the legs, it leads to 

Pythagorean Identity #2 and  #3. 

#2.cos^2θ+sin^2θ = 1        

cos^2θ  cos^2θ  cos^2θ
1+tan^2θ=sec^2θ

#3.cos^2θ+sin^2θ = 1        

sin^2θ   sin^2θ     sin^2θ
cot^2θ+1=csc^2θ