RIGHT TRIANGLES AND TRIGONOMETRY

By Brianna Meikle

The Pythagorean Theorem

Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

o c²=a²+b² c is always the hypotenuse or the

diagonal, while a and b are the legs.

Finding the Length of a Hypotenuse

(hypotenuse)²=(leg)²+(leg)²

x²=3²+4²

x²=9+16

x²=25

x=5

Pythagorean Theorem

Substitute

Multiply

Add

Find the positive square root

Example of finding Hypotenuse

What you’re doing

Finding the Length of a Leg

Let c=13² and b=5²13²=5²+a²169=25+ a²144=a²a=12

Theorems About Special Right Triangles

45°-45°-90° Triangle Theorem

In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.

30°-60°-90° Triangle Theorem

In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

Finding a leg in a 45-45-90 triangle and the side lengths in a 30-60-90 triangle.

Finding a leg in a 45-45-90 Triangle

Longer Leg in a 30-60-90 Triangle

What to do:45-45-90 Triangle Theorem

30-60-90 Triangle Theorem

Substitute SubstituteDivide each side by √2. Divide each side by √3.Simplify Simplify* Numerator and Denominator by √2.

*Numerator and Denominator by √3.

Simplify Simplify

Finding Trigonometric Ratios

S.O.H.C.A.H.T.O.A.These letters stand for:Sine, Cosine, and Tangent To find the sine A:Use the Opposite Side over the Hypotenuse.To find the cosine A:Use the Adjacent Side over the Hypotenuse.To find the tangent A:Use the Opposite Side over the Adjacent Side.

Learning the sine, cosine, and tangent will give you the ability to find the lengths of triangles.

What is the necessity for the sine, cosine, and tangent?

How well did Students Understand this chapter?

90%

100%40%

30%

95%

Understanding Chapter 9 Student #1

Student #2Student #3Student #4Student #5

What percentage of the time did students understand Chapter Nine? This year or past years

Student #1 90%Student #2 100%Student #3 40%Student #4 30%Student #5 95%