Trigonometry

Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit circle 5. Inverse trig functions 1. Exact values 3. Changing units. 5. Calculator work 1. Law of Sines 2. Law of Cosines : AAS, ASA, SSA : SAS, SSS

Right Triangles “naming the sides of the triangle.” What we call the legs of the triangle depend on the non- right angle given. hypotenuse opposite adjacent opposite adjacent This is important because all of the trig functions are ratios that are defined by the lengths of these sides. For example: sine of an angle is the ratio of the length of the side opposite the angle divided by the length of the hypotenuse. Sin θ =

Confused? hypotenuse opposite adjacent

hypotenuse opposite adjacent

Trig Functions There are 6 trig functions we must be able to use. We must memorize their EXACT values in both radical and radian form. Remember: trig functions are the result of ratios of the lengths of sides of a right triangle.

Trig Functions There are 3 main trig functions and the 3 that are reciprocals of the first three. The main ones are: Sine, Cosine and Tangent. sin θ = opp hyp cos θ = adj hyp tan θ = opp adj The reciprocals are: Cosecant, Secant and Cotangent. csc θ = hyp opp sec θ = hyp adj cot θ = adj opp Basically, to find the trig relationship of any angle on a right triangle, all we need to do is measure the appropriate sides of that triangle. This is called “evaluating the trig functions of an angle θ.”

hypotenuse opposite adjacent Evaluate the six trig functions of the angle θ. θ sin θ = opp hyp sin θ = 3 5 cos θ = adj hyp cos θ = 4 5 tan θ = opp adj tan θ = 3 4 csc θ = 5 3 sec θ = 5 4 cot θ = 4 3

hypotenuse opposite adjacent We can work backwards as well. If they give us the ratio, we can find the other trig functions. θ sin θ = opp hyp Given: sin θ = 5 6 cos θ = 6 tan θ = 5 sec θ = 6 csc θ = 6 5 cot θ = a 2 + b 2 = c 2 sec θ = 6 11 tan θ = 5 11

Special Triangles: and ˚ 60˚ 45˚

30˚ 60˚ 45˚ θsin θcos θtan θcsc θsec θtan θ 30˚ 60˚ 45˚ θsin θcos θtan θcsc θsec θtan θ

Find the exact values of x and y. 60˚ x 8 y

Find the values of x and y. 35˚ y x 16

This is 1 unit long. 180˚ = π radians 360˚ = 2π radians Hence the name: The UNIT CIRCLE 90˚ = π radians 2

Since 180 ˚ = 1π radians we can us this as our conversion factor. In other words to change degrees into radian we multiply by π 180˚ To change radians into degrees we multiply by π 180˚ Hint: What we “want” is always in the numerator. If we want our final answer in degrees then 180 ˚ is on top. If we want radians then π radians in on top!

Convert 230˚ to radians. Since we want radians we multiply by π/18 (radians in the numerator. 230˚ ● π = 230π 180˚ 180 Which reduces to 23π 18 NO MIXED FRACTIONS!!!

Convert π to degrees 12 Since we want degrees we multiply by 180/π (degrees in the numerator.) Notice the π’s cancel! Reduces to 15˚

● ● ● ● (4, 12) adjacent opposite hypotenuse radius This leads us to believe that there must be a connection between sin, cos and the coordinates (x, y)

The UNIT CIRCLE Remember the unit circle has a radius of 1 unit. ● So to find the coordinates of this point we can use the sin and cos if we know what the measure of the angle formed by the radius and the x axis is.. θ ( the length of the pink line, the length of the red line) BUT WAIT! That’s what cos and sin are defined as! sin θ = length of side opposite length of hypotenuse cosθ = length of side adjacent length of hypotenuse AND WE KNOW THAT THE RADIUS IN A UNIT CIRCLE IS 1 so that means: sin θ = length of side opposite cosθ = length of side adjacent ( cos θ, sin θ )

What are the coordinates of ● θ ( the length of the pink line, the length of the red line) ( cos θ, sin θ )

The UNIT CIRCLE ● ● ● ● (4, 12) adjacent opposite hypotenuse radius