Now That We Know Angles and Radians, We Can Change Our Focus to Triangles

4.3

Now that we know angles and radians, we can change our focus to triangles.

Why would we ever need to know about triangles?

Say, for example, that we are going to put up a basketball hoop in our driveway. We need to know the height of the hoop and how far away the foul line should be.

Review of Triangles.

x = opposite leg y = adjacent leg z = hypotenuse

At this point, we are going to learn a word that will stay with us for the remainder of the year. On the last day of the year, we will all still remember this word and exactly what it stands for.

S O H C A H T O A

S in = O pposite H ypotenuse

C osine = A djacent H ypotenuse

T angent = O pposite A djacent

Getting back to our triangle, find the Sine, Cosine, and Tangent of our angle

Sin Cos Tan

Getting back to our basketball hoop, we know that the height of the hoop is supposed to be 10 ft and that the angle of elevation from the foul line to the hoop is 33.7 degrees. We want to use this information to figure out how far away the foul line should be.

Tan 33.7 = x = x = 15 ft

Suppose you are standing parallel to the Concord River. You turn and walk at a 60 degree angle until you reach the river, which is a distance of 130 ft. How much farther down the river are you than when you started?

You are playing paintball with a friend when you notice him sitting in a tree. The angle of elevation to your friend is 23 degrees. If the distance between yourself and the tree is 60 ft, how high up in the tree is your friend?

You are a surveyor standing 115 ft from the Washington Monument. You measure the angle of elevation to the top of the monument to be 78.3 degrees. How tall is the Washington Monument?

We can also use our trigonometric functions to find angles if we are given different sides. For example, lets take a look at a triangle with legs 5, 8, and a hypotenuse of 15. We can use our trig functions to find all the angles in this triangle.

Add in 2 or 3 examples on finding angles when given sides.

2 Special Types of Triangles

1) 30 - 60 - 90 Triangle

What is an equilateral triangle?

if all 3 sides are the same, then all 3 angles are the same. Thus, each angle would be 60 degrees.

Here, use this to introduce a 30 - 60 - 90 triangle

2) 45 - 45 - 90 Triangle

A slide has a 12 ft ladder which descends at a 45 degree angle. How long is the slide and how far from the ladder is the edge of the slide. Give EXACT values.

A ladder leans against the side of a house making a 60 degree angle with the ground. If the ladder is 10 ft long, how far up the house does it reach?

On page 283 of our textbooks, there are 3 extremely important identities:

Pythagorean Identities

1) Sin + Cos = 1

2) 1 + Tan = Sec

3) 1 + Cot = Csc

Also, we have the 2 identities:

Tan and Cot

We are told that the Sin = 0.6 and we are to find the Cos and the Tan . How can we do this?

- from our first property, we can replace Sin with 0.6 to solve for the Cos

Cos = 0.8

- from our last 2 identities, we can find Tan by substituting in 0.6 for the Sin and 0.8 for the Cos . By dividing this two, we can find the Tan .

Tan = 0.75

The Cot = ¼

Find the Tan and the Sec

We know that the Cot =

From this, we can deduce that the Tan = 4

From here, we can use our identities to find the Sec .

Solve the following for x.

1) Sin - x = - Cos

2) (1 + Sin ) (1 - Sin ) = x

3) Tan = x Sec

4) Csc - Cot = x

5) Tan - Sec = x

6) 2 Sin + Cos - Sin + 1 = x