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MPM2D: Principles of Mathematics

Right-Angled TrigonometryApplications

J. Garvin

Slide 1/10

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Using Trigonometric Ratios

Recap

Determine the measure of ∠A.

cosA = 1623

A = cos−1(1623

)A ≈ 46◦

J. Garvin — Right-Angled Trigonometry

Slide 2/10

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Using Trigonometric Ratios

Recap

Determine the measure of ∠A.

cosA = 1623

A = cos−1(1623

)A ≈ 46◦

J. Garvin — Right-Angled Trigonometry

Slide 2/10

tr i gonometry

Applications of Trigonometric Ratios

Recall all of the tools available for use with right triangles:

• Pythagorean Theorem (find a side, given two sides)

• trigonometric ratios (find a side, given a side and anangle)

• inverse trigonometric ratios (find an angle, given twosides)

Sometimes there is more than one way to solve a problem,but usually one method is more efficient than another.

J. Garvin — Right-Angled Trigonometry

Slide 3/10

tr i gonometry

Applications of Trigonometric Ratios

Recall all of the tools available for use with right triangles:

• Pythagorean Theorem (find a side, given two sides)

• trigonometric ratios (find a side, given a side and anangle)

• inverse trigonometric ratios (find an angle, given twosides)

Sometimes there is more than one way to solve a problem,but usually one method is more efficient than another.

J. Garvin — Right-Angled Trigonometry

Slide 3/10

tr i gonometry

Applications of Trigonometric Ratios

Recall all of the tools available for use with right triangles:

• Pythagorean Theorem (find a side, given two sides)

• trigonometric ratios (find a side, given a side and anangle)

• inverse trigonometric ratios (find an angle, given twosides)

Sometimes there is more than one way to solve a problem,but usually one method is more efficient than another.

J. Garvin — Right-Angled Trigonometry

Slide 3/10

tr i gonometry

Applications of Trigonometric Ratios

Recall all of the tools available for use with right triangles:

• Pythagorean Theorem (find a side, given two sides)

• trigonometric ratios (find a side, given a side and anangle)

• inverse trigonometric ratios (find an angle, given twosides)

Sometimes there is more than one way to solve a problem,but usually one method is more efficient than another.

J. Garvin — Right-Angled Trigonometry

Slide 3/10

tr i gonometry

Applications of Trigonometric Ratios

Recall all of the tools available for use with right triangles:

• Pythagorean Theorem (find a side, given two sides)

• trigonometric ratios (find a side, given a side and anangle)

• inverse trigonometric ratios (find an angle, given twosides)

Sometimes there is more than one way to solve a problem,but usually one method is more efficient than another.

J. Garvin — Right-Angled Trigonometry

Slide 3/10

tr i gonometry

Applications of Trigonometric Ratios

Example

A wire, attached 4 m up a telephone pole, makes an angle of72◦ with the ground. How long is the wire?

In the diagram, w is thelength of the wire.

sin 72◦ =4

w

w =4

sin 72◦

w ≈ 4.2 m

J. Garvin — Right-Angled Trigonometry

Slide 4/10

tr i gonometry

Applications of Trigonometric Ratios

Example

A wire, attached 4 m up a telephone pole, makes an angle of72◦ with the ground. How long is the wire?

In the diagram, w is thelength of the wire.

sin 72◦ =4

w

w =4

sin 72◦

w ≈ 4.2 m

J. Garvin — Right-Angled Trigonometry

Slide 4/10

tr i gonometry

Applications of Trigonometric Ratios

Example

A wire, attached 4 m up a telephone pole, makes an angle of72◦ with the ground. How long is the wire?

In the diagram, w is thelength of the wire.

sin 72◦ =4

w

w =4

sin 72◦

w ≈ 4.2 m

J. Garvin — Right-Angled Trigonometry

Slide 4/10

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Applications of Trigonometric Ratios

Sometimes an angle is specified in relation to another object,such as “10◦ with the ground” or “30◦ to the wall”.

In other cases, an angle may be specified relative to ahorizontal line. An angle measured upward from this line isan angle of elevation, while an angle measured downward isan angle of depression.

J. Garvin — Right-Angled Trigonometry

Slide 5/10

tr i gonometry

Applications of Trigonometric Ratios

Sometimes an angle is specified in relation to another object,such as “10◦ with the ground” or “30◦ to the wall”.

In other cases, an angle may be specified relative to ahorizontal line. An angle measured upward from this line isan angle of elevation, while an angle measured downward isan angle of depression.

J. Garvin — Right-Angled Trigonometry

Slide 5/10

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Applications of Trigonometric Ratios

Example

From atop of a 35 m vertical cliff, the angle of depression toa large rock below is 61◦. How far from the base of the cliffis the rock?

In the diagram, d is thedistance to the rock.Note that the angle ofdepression falls outside of thetriangle, creating a 29◦ angleinside.Also note that the angle ofelevation from the rock to thecliff top is 61◦, due to interioralternate angles.

J. Garvin — Right-Angled Trigonometry

Slide 6/10

tr i gonometry

Applications of Trigonometric Ratios

Example

From atop of a 35 m vertical cliff, the angle of depression toa large rock below is 61◦. How far from the base of the cliffis the rock?

In the diagram, d is thedistance to the rock.

Note that the angle ofdepression falls outside of thetriangle, creating a 29◦ angleinside.Also note that the angle ofelevation from the rock to thecliff top is 61◦, due to interioralternate angles.

J. Garvin — Right-Angled Trigonometry

Slide 6/10

tr i gonometry

Applications of Trigonometric Ratios

Example

From atop of a 35 m vertical cliff, the angle of depression toa large rock below is 61◦. How far from the base of the cliffis the rock?

In the diagram, d is thedistance to the rock.Note that the angle ofdepression falls outside of thetriangle, creating a 29◦ angleinside.

Also note that the angle ofelevation from the rock to thecliff top is 61◦, due to interioralternate angles.

J. Garvin — Right-Angled Trigonometry

Slide 6/10

tr i gonometry

Applications of Trigonometric Ratios

Example

From atop of a 35 m vertical cliff, the angle of depression toa large rock below is 61◦. How far from the base of the cliffis the rock?

In the diagram, d is thedistance to the rock.Note that the angle ofdepression falls outside of thetriangle, creating a 29◦ angleinside.Also note that the angle ofelevation from the rock to thecliff top is 61◦, due to interioralternate angles.

J. Garvin — Right-Angled Trigonometry

Slide 6/10

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Applications of Trigonometric Ratios

Using either angle should produce the same answer, but willchange the setup of the ratio.

Using the 29◦ angle. . .

tan 29◦ =d

35d = 35 × tan 29◦

d ≈ 19.4 m

Using the 61◦ angle. . .

tan 61◦ =35

d

d =35

tan 61◦

d ≈ 19.4 m

Using either method, the rock is approximately 19.4 m fromthe cliff.

J. Garvin — Right-Angled Trigonometry

Slide 7/10

tr i gonometry

Applications of Trigonometric Ratios

Using either angle should produce the same answer, but willchange the setup of the ratio.

Using the 29◦ angle. . .

tan 29◦ =d

35d = 35 × tan 29◦

d ≈ 19.4 m

Using the 61◦ angle. . .

tan 61◦ =35

d

d =35

tan 61◦

d ≈ 19.4 m

Using either method, the rock is approximately 19.4 m fromthe cliff.

J. Garvin — Right-Angled Trigonometry

Slide 7/10

tr i gonometry

Applications of Trigonometric Ratios

Using either angle should produce the same answer, but willchange the setup of the ratio.

Using the 29◦ angle. . .

tan 29◦ =d

35d = 35 × tan 29◦

d ≈ 19.4 m

Using the 61◦ angle. . .

tan 61◦ =35

d

d =35

tan 61◦

d ≈ 19.4 m

Using either method, the rock is approximately 19.4 m fromthe cliff.

J. Garvin — Right-Angled Trigonometry

Slide 7/10

tr i gonometry

Applications of Trigonometric Ratios

Using either angle should produce the same answer, but willchange the setup of the ratio.

Using the 29◦ angle. . .

tan 29◦ =d

35d = 35 × tan 29◦

d ≈ 19.4 m

Using the 61◦ angle. . .

tan 61◦ =35

d

d =35

tan 61◦

d ≈ 19.4 m

Using either method, the rock is approximately 19.4 m fromthe cliff.

J. Garvin — Right-Angled Trigonometry

Slide 7/10

tr i gonometry

Applications of Trigonometric Ratios

Example

The base of a 4.8 m ladder is placed 1.5 m from a wall. Forsafety reasons, the angle of elevation cannot exceed 67◦. Isthe ladder safe?

In the diagram, x is the angleof elevation.

cos x =1.5

4.8

x = cos−1

(1.5

4.8

)x ≈ 72◦

Since 72◦ > 67◦, the ladder isnot safe.

J. Garvin — Right-Angled Trigonometry

Slide 8/10

tr i gonometry

Applications of Trigonometric Ratios

Example

The base of a 4.8 m ladder is placed 1.5 m from a wall. Forsafety reasons, the angle of elevation cannot exceed 67◦. Isthe ladder safe?

In the diagram, x is the angleof elevation.

cos x =1.5

4.8

x = cos−1

(1.5

4.8

)x ≈ 72◦

Since 72◦ > 67◦, the ladder isnot safe.

J. Garvin — Right-Angled Trigonometry

Slide 8/10

tr i gonometry

Applications of Trigonometric Ratios

Example

The base of a 4.8 m ladder is placed 1.5 m from a wall. Forsafety reasons, the angle of elevation cannot exceed 67◦. Isthe ladder safe?

In the diagram, x is the angleof elevation.

cos x =1.5

4.8

x = cos−1

(1.5

4.8

)x ≈ 72◦

Since 72◦ > 67◦, the ladder isnot safe.

J. Garvin — Right-Angled Trigonometry

Slide 8/10

tr i gonometry

Applications of Trigonometric Ratios

Example

The base of a 4.8 m ladder is placed 1.5 m from a wall. Forsafety reasons, the angle of elevation cannot exceed 67◦. Isthe ladder safe?

In the diagram, x is the angleof elevation.

cos x =1.5

4.8

x = cos−1

(1.5

4.8

)x ≈ 72◦

Since 72◦ > 67◦, the ladder isnot safe.

J. Garvin — Right-Angled Trigonometry

Slide 8/10

tr i gonometry

Applications of Trigonometric Ratios

Example

Bert lives due east of school. Ernie lives due south of school,and is four times as far from school as Bert. If Bert and Ernielive 3 km apart, how far from school does each person live?

Let b be Bert’s distancefrom school, and 4bErnie’s distance.

b2 + (4b)2 = 32

17b2 = 9

b2 = 917

b =√

917

b ≈ 0.73 km

Bert is about 0.73 kmfrom school, while Ernieis 4 × 0.73 ≈ 2.92 km.

J. Garvin — Right-Angled Trigonometry

Slide 9/10

tr i gonometry

Applications of Trigonometric Ratios

Example

Bert lives due east of school. Ernie lives due south of school,and is four times as far from school as Bert. If Bert and Ernielive 3 km apart, how far from school does each person live?

Let b be Bert’s distancefrom school, and 4bErnie’s distance.

b2 + (4b)2 = 32

17b2 = 9

b2 = 917

b =√

917

b ≈ 0.73 km

Bert is about 0.73 kmfrom school, while Ernieis 4 × 0.73 ≈ 2.92 km.

J. Garvin — Right-Angled Trigonometry

Slide 9/10

tr i gonometry

Applications of Trigonometric Ratios

Example

Bert lives due east of school. Ernie lives due south of school,and is four times as far from school as Bert. If Bert and Ernielive 3 km apart, how far from school does each person live?

Let b be Bert’s distancefrom school, and 4bErnie’s distance.

b2 + (4b)2 = 32

17b2 = 9

b2 = 917

b =√

917

b ≈ 0.73 km

Bert is about 0.73 kmfrom school, while Ernieis 4 × 0.73 ≈ 2.92 km.

J. Garvin — Right-Angled Trigonometry

Slide 9/10

tr i gonometry

Applications of Trigonometric Ratios

Example

Bert lives due east of school. Ernie lives due south of school,and is four times as far from school as Bert. If Bert and Ernielive 3 km apart, how far from school does each person live?

Let b be Bert’s distancefrom school, and 4bErnie’s distance.

b2 + (4b)2 = 32

17b2 = 9

b2 = 917

b =√

917

b ≈ 0.73 km

Bert is about 0.73 kmfrom school, while Ernieis 4 × 0.73 ≈ 2.92 km.J. Garvin — Right-Angled Trigonometry

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Questions?

J. Garvin — Right-Angled Trigonometry

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