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 Hello Guys, so far we have been learning about trigonometric identities in class. This post is about simplifying a trigonometric expression and proving a trigonometric equation using trigonometric identities.

Trigonometric  Identities: A trigonometric equation which is true for all permissible values of the variable for which both sides of the equation are defined is called Trigonometric Identity.

Example: There are:

                                

                        Reciprocal Identities

                              

               
                                   


There are sum and difference identities of sine, cosine, and tangent functions which can be used to simplify an expression or to prove a trigonometric equation.

Example 1:Simplify the expression using trigonometric identities.
                                
                  Rewrite tan as sin/cos.
                              
                              
                              
                              

Example 2:
                         

Graphing Sine and Cosine Functions

Graphing Sine and Cosine Functions

When you graph onto a Cartesian plane, it refers to the "unrolling" of the unit circle.

When graphing, you plot the values of theta onto the x-axis and the trigonometric function values at theta onto the y-axis.

One cycle refers to the part of the graph from one point to another where the graph begins to repeat itself.

One period is how long one cycle is in either degrees or radians. The formula to determine the period of a sin x function or cos x function is 2π/|b| The formula to determine the period of a tan x function is π/|b|

Amplitude is the distance between the middle axis to the highest or lowest point of a sin x function or cos x function. When there is a change in amplitude it vertically stretches or compresses the original shape of the graph. The formula to determine the amplitude of a sin x function or a cos x function is |a|. Whereas the amplitude of a tan x function is infinite.  

Note: amplitude = max - min / 2

Basic Equations:

y = asinbx 

y = acosbx

y = atanbx

x represents theta values

y represents trigonometric function's value at theta

Remember to use quadrantal values for x when you are graphing a sin x function or a cos x function.

A tan x function will have asymptotes at quadrantal values where tan x is undefined.

For example:

Note: When graphing sine functions, you always start at (0,0) to begin with. 

y = sinx

Period (2π/|b|, b =1): 2π/|1| = 1

Amplitude (|a|):  1

Domain: {θ,θER}

Range: [-1, 1]

Maximum Value: @ y = 1

Minimum Value: @ y = - 1

y - intercept: @ y = 0

θ - intercepts:  θ = πn, nEI 

Note: When graphing cosine functions, you always start at (0, |a|) to begin with.

y = cosx

Period (2π/|b|, b =1): 2π/|1| = 1

Amplitude (|a|):  1

Domain: {θ,θER}

Range: [-1, 1]

Maximum Value: @ y = 1

Minimum Value: @ y = - 1

y - intercept: @ y = 1

θ - intercepts:  θ = π/2 + πn, nEI 

Approximate Values

Approximate Values of Trigonometric Ratios 

To find approximate values for sine, cosine, or tangent, we can use our scientific calculators. Before you start calculating, make sure your calculator is set to the right mode. (Radians or Degrees)

Here are some examples:

• sin 4.2  = -0.871575772

To get this value- 1) Set calculator to radians mode
                             2) Punch in sin 4.2

• cos 260° = -0.173648177

To get this value- 1) Set calculator to degrees mode
                            2) Punch in cos 260



You can also find approximate values for cosecant, secant and cotangent. To do this, make sure your calculator is still set to the right mode and use the correct reciprocal function

Examples: 

• sec3.3 = -1.012678974

Since secant is H/A (meaning its the opposite of cos which is A/H) that means you have to divide 1/cos3.3 to find secant 3.3

• cot3 = -7.015252551

Since cotangent is A/O (meaning its the opposite of tan which is O/A) that means you have to divide 1/tan3 to find cotangent 3

Approximate Values of Angles

If you know the value of the trigonometric ratio, you can use the inverse function key on your calculator.

REMEMBER: that sinˆ-1, cosˆ-1, tanˆ-1 means the inverse of sin, the inverse of cos and the inverse of tan NOT (sin30°)ˆ-1, which means the reciprocal of sin30° or 1/sin30°

Examples:

sinϴ= 0.879 in the domain 0≤ϴ≤2π. give answers to the nearest tenth of a radian.



Step 1) Since the question states to give answers in radians, make sure your calculator is set to that mode and if making a diagram, to label in radians


Step 2) Looking at the value of the trig ratio (sin= 0.879) you can state that the solutions are in Quadrants 1 and 2 because that is where sin is positive.

Step 3) You need to find the reference angle, to do this you calculate the inverse of the trig ratio value.  So in this case it would be sinˆ-1(0.879). The reference angle would be 1.073760909

Step 4) Find the measures of the angles. Since you already figured out its in Quadrant 1 and 2, you use the quadrantal angles to help you.

Quadrant 1- ϴ would just equal the reference angle
                    ϴ = 1.073760909
                    ϴ = 1.1 radians

Quadrant 2- ϴ = π- reference angle
                    ϴ = 2.067831744
                    ϴ = 2.1 radians

Step 5) Since both answers fit in the domain, you can accept both.

Unit Circle

Hey Guys!

A few days before we learned about the Unit Circle. A Unit Circle is the circle with its centre at the origin and with a radius of unit

Equation: x^2 + y^2 = 1

Positive distance is measured in  a counterclockwise direction; negative distance is measured in a clockwise direction

The notation P(θ) is used to detonate the terminal point, where the terminal arm of angle θ intercepts the unit circle. For every arc length θ on the unit circle, P(θ) is unique.

Here's a picture of the unit circle in:                                                                                                   

Degrees

                                                           

           

                

Radiant

We can define P(θ) as the ordered pair P(x,y)

Remember, if r=1:

sin θ  =  y
               r       
csc θ  =  r
                y  
cos θ  =  x
               r       
sec θ  =  r
               x  
tan θ  =  y
              x       
cot θ  =  x
               y

or if its a special triangle we use this 

Angle	Sin	Cos	Tan=Sin/Cos
30°			1√3 = √33
45°			1
60°			√3

Note: The equation for a circle  with centre at origin and a radius other than 1 would be x^2 + y^2 = r^2 or cos^2θ + sin^2θ = 1 

Overall look on the unit circle:

Remember:

Good luck everybody on the test! 

Degree and Radian Measure

Angle is determined by two rays, the terminal and initial ray. Angle is represented by a Greek letter theta ϴ. It can be measured using different units, such as revolutions, rotations, degrees, radians, and gradians.

Angle of rotation is defined by rotating a ray about its endpoint, making a vertex. The starting point is the Initial side and the result is called the terminal side.

Angle measures without units are considered in radians. Angles in degrees must show the symbol. Obtuse angles are greater than 90 degrees and acute are less. A positive angle rotates counter clockwise and a angle to be negative, the rotation is clockwise

An angle in standard position must have the vertex as the origin and initial side on the positive x-axis. The measure between the terminal and x-axis is called the reference angle.

Radian measure is another way to measure angles.

Arc length of a circle is found with the  formula a=ϴr. R is the radius and theta is the angle in radians. To
convert to radians you multiply the degrees measure by /180. And to convert degrees to radians you multiply by 180.