4.3 – Right Triangle Trigonometry

Accelerated Pre-Calculus

Mr. Niedert

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 1 / 22

4.3 – Right Triangle Trigonometry

1 The Six Trigonometric Functions

2 Trigonometric Identities

3 Evaluating Trigonometric Functions with a Calculator

4 Applications Involving Right Triangles

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22

4.3 – Right Triangle Trigonometry

1 The Six Trigonometric Functions

2 Trigonometric Identities

3 Evaluating Trigonometric Functions with a Calculator

4 Applications Involving Right Triangles

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22

4.3 – Right Triangle Trigonometry

1 The Six Trigonometric Functions

2 Trigonometric Identities

3 Evaluating Trigonometric Functions with a Calculator

4 Applications Involving Right Triangles

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22

4.3 – Right Triangle Trigonometry

1 The Six Trigonometric Functions

2 Trigonometric Identities

3 Evaluating Trigonometric Functions with a Calculator

4 Applications Involving Right Triangles

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22

Today’s Learning Target(s)

1 I can find the values of all six trigonometric functions for any righttriangle.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 3 / 22

The Six Trigonometric Functions

Consider the triangle below for the definitions of the six trigonometricfunctions.

Right Triangle Definitions of Trigonometric Functions

sin θ =opp

hypcos θ =

adj

hyptan θ =

opp

adj

csc θ =hyp

oppsec θ =

hyp

adjcot θ =

adj

opp

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 4 / 22

The Six Trigonometric Functions

Consider the triangle below for the definitions of the six trigonometricfunctions.

Right Triangle Definitions of Trigonometric Functions

sin θ =opp

hypcos θ =

adj

hyptan θ =

opp

adj

csc θ =hyp

oppsec θ =

hyp

adjcot θ =

adj

opp

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 4 / 22

Evaluating Trigonometric Functions

Practice

In the triangle below, find the values of the six trigonometric functions ofθ.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 5 / 22

Complementary Angles

There is a relationship between cofunctions that will help you inevaluating trigonometric functions.

Sine and cosine are said to be cofunctions. Tangent and cotangentare cofunctions. In addition, secant and cosecant are cofunctions.

This yields the following relationships between the angles.

Cofunctions are Complementary

sin(90◦ − θ) = cos θ tan(90◦ − θ) = cot θ sec(90◦ − θ) = csc θcos(90◦ − θ) = sin θ cot(90◦ − θ) = tan θ csc(90◦ − θ) = sec θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22

Complementary Angles

There is a relationship between cofunctions that will help you inevaluating trigonometric functions.

Sine and cosine are said to be cofunctions. Tangent and cotangentare cofunctions. In addition, secant and cosecant are cofunctions.

This yields the following relationships between the angles.

Cofunctions are Complementary

sin(90◦ − θ) = cos θ tan(90◦ − θ) = cot θ sec(90◦ − θ) = csc θcos(90◦ − θ) = sin θ cot(90◦ − θ) = tan θ csc(90◦ − θ) = sec θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22

Complementary Angles

There is a relationship between cofunctions that will help you inevaluating trigonometric functions.

Sine and cosine are said to be cofunctions. Tangent and cotangentare cofunctions. In addition, secant and cosecant are cofunctions.

This yields the following relationships between the angles.

Cofunctions are Complementary

sin(90◦ − θ) = cos θ tan(90◦ − θ) = cot θ sec(90◦ − θ) = csc θcos(90◦ − θ) = sin θ cot(90◦ − θ) = tan θ csc(90◦ − θ) = sec θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22

Complementary Angles

There is a relationship between cofunctions that will help you inevaluating trigonometric functions.

Sine and cosine are said to be cofunctions. Tangent and cotangentare cofunctions. In addition, secant and cosecant are cofunctions.

This yields the following relationships between the angles.

Cofunctions are Complementary

sin(90◦ − θ) = cos θ tan(90◦ − θ) = cot θ sec(90◦ − θ) = csc θcos(90◦ − θ) = sin θ cot(90◦ − θ) = tan θ csc(90◦ − θ) = sec θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22

4.3 – Right Triangle Trigonometry (Part 1 of 3)Assignment

Part 1: pg. 308 #1-4, 6-26 even

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 7 / 22

Today’s Learning Target(s)

1 I can apply the trigonometric identities to find the values of varioustrigonometric functions.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 8 / 22

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities

Reciprocal Identities

sin θ =1

csc θcos θ =

1

sec θtan θ =

1

cot θ

csc θ =1

sin θsec θ =

1

cos θcot θ =

1

tan θ

Quotient Identities

tan θ =sin θ

cos θcot θ =

cos θ

sin θ

Pythagorean Identities

sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ1 + cot2 θ = csc2 θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 9 / 22

Applying Trigonometric Identities

Example

Let θ be an acute angle such that sin θ = 0.6. Find the values of (a) cos θand (b) tan θ using trigonometric identities.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 10 / 22

Applying Trigonometric Identities

Practice

Let θ be an acute angle such that cos θ = 0.96. Find the values of (a)sin θ and (b) tan θ using trigonometric identities.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 11 / 22

Applying Trigonometric Identities

Example

Let θ be an acute angle such that tan θ = 3. Find the values of (a) cot θand (b) sec θ using trigonometric identities.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 12 / 22

Applying Trigonometric Identities

Practice

Let β be an acute angle such that tanβ = 4. Find the values of (a) cotβand (b) secβ using trigonometric identities.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 13 / 22

Transforming Trigonometric Identities

Practice

Use trigonometric identities to transform the left side of the equation intothe right side.

a tan θ cot θ = 1

b (1 + cos θ) (1 − cos θ) = sin2 θ

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 14 / 22

4.3 – Right Triangle Trigonometry (Part 2 of 3)Assignment

Part 1: pg. 308 #1-4, 6-26 evenPart 2: pg. 309 #28-40 even, 41-42

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 15 / 22

Today’s Learning Target(s)

1 I can evaluate trigonometric functions using a calculator in bothradians and degrees.

2 I can apply trigonometric functions to solve angles of elevation andangle of depression problems.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 16 / 22

Using a Calculator

Example

Use a calculator to evaluate each of the following.

a cos 28◦

b sec 32◦15′32′′

c tan 5π12

d cot 1

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 17 / 22

Using a Calculator

Practice

Use a calculator to evaluate each of the following.

a sin 49◦

b csc 72◦35′49′′

c cos 1π5

d sec 1

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 18 / 22

Using Trigonometry to Solve a Right Triangle

Practice

A surveyor is standing 115 feet from the base of the WashingtonMonument, as shown below. The surveyor measure the angle of elevationto the top of the monument as 78.3◦. How tall is the WashingtonMonument?

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 19 / 22

Using Trigonometry to Solve a Right Triangle

Practice

A historic lighthouse is 200 yards from a bike path along the edge of alake. A walkway to the lighthouse is 400 yards long. Find the acute angleθ between the bike path and the walkway.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 20 / 22

Solving a Right Triangle

Practice

Find the length c of the skateboard ramp below.

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 21 / 22

4.3 – Right Triangle Trigonometry (Part 3 of 3)Assignment

Part 1: pg. 308 #1-4, 6-26 evenPart 2: pg. 309 #28-40 even, 41-42Part 3: pg. 309-310 #44-52 even, 59-68

4.3 – Right Triangle Trigonometry Assignmentpg. 308-310 #1-4, 6-26 even, 28-40 even, 41-42, 44-52 even, 59-68

Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 22 / 22