Trigonometry

Trigonometry Non-Right Triangles Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 1. Exact values 1. Law of Sines : AAS, ASA, SSA 2. Law of Cosines : SAS, SSS 2. a2 + b2 = c2 3. Radian Measure of angles 3. Changing units. 4. Unit circle 5. Inverse trig functions 5. Calculator work

“naming the sides of the triangle.” Right Triangles “naming the sides of the triangle.” What we call the legs of the triangle depend on the non-right angle given. Sin θ = opposite adjacent hypotenuse 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.

Confused? adjacent hypotenuse opposite

opposite hypotenuse 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 reciprocals are: Cosecant, Secant and Cotangent. The main ones are: Sine, Cosine and Tangent. sin θ = opp hyp csc θ = hyp 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. cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp This is called “evaluating the trig functions of an angle θ.”

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

We can work backwards as well We can work backwards as well. If they give us the ratio, we can find the other trig functions. Given: sin θ = 5 6 θ 6 adjacent hypotenuse sin θ = opp hyp a2 + b2 = c2 csc θ = 6 5 cos θ = 6 sec θ = 6 opposite 5 sec θ = 6 11 tan θ = 5 11 tan θ = 5 cot θ = 5

Special Triangles: 30-60-90 and 45-45-90 30˚ 45˚ 2 1 60˚ 45˚ 1 1

θ sin θ cos θ tan θ csc θ sec θ tan θ θ sin θ cos θ tan θ csc θ sec θ 30˚ 45˚ 60˚ θ sin θ cos θ tan θ csc θ sec θ tan θ 30˚ 45˚ 60˚

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

Find the values of x and y. 16 35˚ y

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

Since 180 ˚ = 1π radians we can us this as our conversion factor. 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! In other words to change degrees into radian we multiply by π To change radians into degrees we multiply by π 180˚ 180˚

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

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

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

Remember the unit circle has a radius of 1 unit. AND WE KNOW THAT THE RADIUS IN A UNIT CIRCLE IS 1 so that means: sinθ = length of side opposite cosθ = length of side adjacent BUT WAIT! That’s what cos and sin are defined as! sinθ = length of side opposite length of hypotenuse cosθ = length of side adjacent 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.. Remember the unit circle has a radius of 1 unit. ( the length of the pink line, the length of the red line) ( cos θ, sin θ ) ● θ

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

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