Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry - - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1)

William (Bill) Finch

Mathematics Department Denton High School

Introduction Trig Ratios Solve Right Triangle Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

• W. Finch

DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

• W. Finch

DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

• W. Finch

DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

• W. Finch

DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Trigonometry

The word trigonometry comes from the Greek language for “measurement of triangles.” The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles.

• W. Finch

DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Trigonometry

The word trigonometry comes from the Greek language for “measurement of triangles.” The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles.

• W. Finch

DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Trigonometry

The word trigonometry comes from the Greek language for “measurement of triangles.” The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles.

• W. Finch

DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Six Trigonometric Ratios

◮ θ (theta) is an acute angle ◮ ‘opp’ is the side opposite to θ ◮ ‘adj’ is the side adjacent to θ ◮ ‘hyp’ is the hypotenuse

adj hyp

• pp

θ sine(θ) = sin θ = opp hyp cosine(θ) = cos θ = adj hyp tangent(θ) = tan θ = opp adj cosecant(θ) = csc θ = hyp

• pp

secant(θ) = sec θ = hyp adj cotangent(θ) = cot θ = adj

• pp
• W. Finch

DHS Math Dept Right Triangle Trig 4 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Six Trigonometric Ratios

◮ θ (theta) is an acute angle ◮ ‘opp’ is the side opposite to θ ◮ ‘adj’ is the side adjacent to θ ◮ ‘hyp’ is the hypotenuse

adj hyp

• pp

θ sine(θ) = sin θ = opp hyp cosine(θ) = cos θ = adj hyp tangent(θ) = tan θ = opp adj cosecant(θ) = csc θ = hyp

• pp

secant(θ) = sec θ = hyp adj cotangent(θ) = cot θ = adj

• pp
• W. Finch

DHS Math Dept Right Triangle Trig 4 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Reciprocal Functions

The cosecant, secant, and cotangent functions are called the reciprocal functions. csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ

• W. Finch

DHS Math Dept Right Triangle Trig 5 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 1

Find exact values for the six trigonometric functions of θ. 7 25 24 θ

• W. Finch

DHS Math Dept Right Triangle Trig 6 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 2

If sin θ = 1 3, find exact values of the five remaining trigonometric functions for the acute angle θ.

• W. Finch

DHS Math Dept Right Triangle Trig 7 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Special Angles

30◦-60◦-90◦ Triangle √ 3x 2x x 30◦ 60◦ 45◦-45◦-90◦ Triangle x x √ 2x 45◦ 45◦ θ 30◦ 45◦ 60◦ sin θ

1 2 √ 2 2 √ 3 2

cos θ

√ 3 2 √ 2 2 1 2

tan θ

√ 3 3

1 √ 3 csc θ 2 √ 2

2 √ 3 3

sec θ

2 √ 3 3

√ 2 2 cot θ √ 3 1

√ 3 3

• W. Finch

DHS Math Dept Right Triangle Trig 8 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Solve a Right Triangle

To solve a right triangle is to find unknown side lengths and/or unknown angles. C b A c B a

• W. Finch

DHS Math Dept Right Triangle Trig 9 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 3

Find the value of x. 7 x 55◦

• W. Finch

DHS Math Dept Right Triangle Trig 10 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Inverse Trigonometric Functions

Inverse Sine If sin θ = x, then sin−1 x = θ. Inverse Cosine If cos θ = x, then cos−1 x = θ. Inverse Tangent If tan θ = x, then tan−1 x = θ.

• W. Finch

DHS Math Dept Right Triangle Trig 11 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 4

Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. 12 15.7 θ

• W. Finch

DHS Math Dept Right Triangle Trig 12 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 5

Solve the right triangle. G h F 28 H f 41.4◦

• W. Finch

DHS Math Dept Right Triangle Trig 13 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 6

Solve the right triangle. B 9 A 5 C a

• W. Finch

DHS Math Dept Right Triangle Trig 14 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Angles of Elevation and Depression

An angle of elevation is the angle formed by a horizontal line and an observer’s line of sight up to an object. Observer Object Elevation An angle of depression is the angle formed by a horizontal line and an observer’s line of sight down to an object below. Observer Object Depression

• W. Finch

DHS Math Dept Right Triangle Trig 15 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 7

Split Rock Lighthouse has stood on the north shore of Lake Superior since 1909. When first lit in 1910 the light could be seen from up to 35 km (a little over 20 miles). The lighthouse is 16 m tall and sits atop a cliff that is 40 m. If a boat was on the lake at a distance of 35 km from the lighthouse, what would be the angle of depression from the top of the lighthouse?

• W. Finch

DHS Math Dept Right Triangle Trig 16 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

Example 8

At a point 300 feet from the base of the CN Tower the angle of elevation up to the SkyPod (once the worlds highest public observation deck) is 78.4◦ and the angle of elevation to the top of the tower is 80.6◦. How much higher above the SkyPod is the top of the tower?

• W. Finch

DHS Math Dept Right Triangle Trig 17 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

◮ Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,

31-35 odd, 39-45 odd, 49, 53

• W. Finch

DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

◮ Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,

31-35 odd, 39-45 odd, 49, 53

• W. Finch

DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

◮ Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,

31-35 odd, 39-45 odd, 49, 53

• W. Finch

DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

◮ Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,

31-35 odd, 39-45 odd, 49, 53

• W. Finch

DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for acute angles of

right triangles.

◮ Solve right triangles. ◮ Apply right triangle trigonometry to model real-world

applications.

◮ Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,

31-35 odd, 39-45 odd, 49, 53

• W. Finch

DHS Math Dept Right Triangle Trig 18 / 18