Right Triangle Trigonometry

23 March 2011

Degree Mode v. Radian Mode

Symbols

Theda – Represents the angle measure

Hypotenuse

Opposite Side

Adjacent

Side

Six Trigonometric Ratios

3 Basic Ratios + 3 Reciprocal Ratios What is a reciprocal?

Six Trigonometric Ratios, cont.

Basic Trig. Ratio Sine Cosine Tangent

Reciprocal Trig. Ratio Cosecant Secant Cotangent

It’s a sin to have two c’s.

Three Basic Trig. Ratios

SOH-CAH-TOA

Sine (SOH)

hypotenuse

oppositesin

24 25

7

Cosine (CAH)

hypotenuse

adjacentcos

24 25

7

Tangent (TOA)

adjacent

oppositetan

24 25

7

Cosecant – Reciprocal of Sine

opposite

hypotenusecsc

24 25

7

sin

1csc

(“It’s a sin to have two C’s.”)

Secant – Reciprocal of Cosine

adjacent

hypotenusesec 24 25

7

cos

1sec

Cotangent – Reciprocal of Tangent

opposite

adjacentcot 24 25

7

tan

1cot

Your Turn:

Pg. 419: 9 – 14, 27 – 32

Solving for Side Lengths

If given one side and one angle measure, then we can solve for any other side of the triangle.

8

x

65

Solving Right Triangles, cont.

1. Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse?

2. Pick the appropriate trig function to solve for x.

3. Solve for x.

8

x

65

Solving for Side Lengths, cont.

8

x

65

Solving Side Lengths, cont.

14x

30

Special Trigonometric RatiosMemorize These!!!

30° 45° 60°

sin

cos

tan

2

1

2

2

2

32

3

2

2

2

1

3

3 1 3

Your Turn:

Inverse Trigonometric Ratios

We can “undo” trig ratios Gives us the angle measurement (theda) Represented by a small –1 in the upper right hand

corner Ex.

2nd button → correct trig ratio

1sin

Inverse Trigonometric Ratios, cont.

6.0cos

Your Turn: Solve for thedaRound to nearest hundredth

5.0sin

5.0cos 2

3sin

1tan

Solving For Angle Measures

If given two sides of a triangle, then we can solve for any of the angles of the triangle.

54

Solving for Angle Measures, cont.

1. Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse?

2. Pick the appropriate trig function to solve for

3. Solve for using the inverse trigonometric function

54

Solving for Angle Measures, cont.

54

Your Turn:

Complete problems 11 – 16 on the Solving Right Triangles Practice handout

Solving Right Triangles

We can use two properties of triangles to solve for all the angles and the side lengths of a right triangle.

Properties of Triangles

Pythagorean Theorem

For a right triangle,

a2 + b2 = c2

Triangle Sum Theorem

When you add up all the angles in a triangle, they equal 180°

Tricks for Solving Right Triangles

Given Two Sides

1. Use Pythagorean Theorem to solve for remaining side.

2. Solve for 1 of the angles using trig ratios

3. Solve for the other angle using Triangle Sum Theorem

Given an Angle & a Side

1. Use the Triangle Sum Theorem to solve for the other angle

2. Use trig ratios to solve for 1 of the sides

3. Use the Pythagorean Theorem to solve for the other side

Beta

– Another symbol for an unknown angle measure

Solving Right Triangles – Examples: Given Two Sides

54

Solving Right Triangles – Examples: Given an Angle and a Side

2

30°