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MPM2D: Principles of Mathematics

Primary Trigonometric Ratios

J. Garvin

Slide 1/10

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Similar Triangles

In the diagram below, ∆ABC ∼ ∆ADE since ∠A is commonto both triangles, and ∠ACB = ∠AED.

This means that any ratio of two sides in ∆ABC is equal tothe ratio of corresponding sides in ∆ADE .

J. Garvin — Primary Trigonometric Ratios

Slide 2/10

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Similar Triangles

In the diagram below, ∆ABC ∼ ∆ADE since ∠A is commonto both triangles, and ∠ACB = ∠AED.

This means that any ratio of two sides in ∆ABC is equal tothe ratio of corresponding sides in ∆ADE .

J. Garvin — Primary Trigonometric Ratios

Slide 2/10

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Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.

J. Garvin — Primary Trigonometric Ratios

Slide 3/10

tr i gonometry

Similar Triangles

By varying the measure of ∠A, the ratio of two sides in∆ABC will change, but will remain equal to the ratio ofcorresponding sides in ∆ADE .

Therefore, a specific measure of ∠A can be associated with aspecific ratio of two sides in a right triangle.

Is the ratio of two sides associated with a given angle unique?

Consider the ratio of the opposite side to the hypotenuse.

If ∠A increases, the length of the opposite side alsoincreases. Thus, the ratio will increase.

If ∠A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.

In both scenarios, the ratio changes with the measure of ∠A.Therefore, the ratio associated with a specific angle is unique.J. Garvin — Primary Trigonometric Ratios

Slide 3/10

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Primary Trigonometric Ratios

In the right triangle ∆ABC below, the three sides have beenlabelled based on their position relative to ∠A.

The opposite and adjacent sides are reversed relative to ∠B,but the hypotenuse is always across from the right angle.

J. Garvin — Primary Trigonometric Ratios

Slide 4/10

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Primary Trigonometric Ratios

In the right triangle ∆ABC below, the three sides have beenlabelled based on their position relative to ∠A.

The opposite and adjacent sides are reversed relative to ∠B,but the hypotenuse is always across from the right angle.

J. Garvin — Primary Trigonometric Ratios

Slide 4/10

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Primary Trigonometric Ratios

There are six possible ratios of sides that can be made fromthe three sides.

The three primary trigonometric ratios are sine, cosine andtangent.

Primary Trigonometric Ratios

Let ∆ABC be a right triangle with ∠A 6= 90◦. Then, thethree primary trigonometric ratios for ∠A are:

Sine: sinA = oppositehypotenuse

Cosine: cosA = adjacenthypotenuse

Tangent: tanA = oppositeadjacent

The phrase SOH-CAH-TOA is a mnemonic for these ratios.

J. Garvin — Primary Trigonometric Ratios

Slide 5/10

tr i gonometry

Primary Trigonometric Ratios

There are six possible ratios of sides that can be made fromthe three sides.

The three primary trigonometric ratios are sine, cosine andtangent.

Primary Trigonometric Ratios

Let ∆ABC be a right triangle with ∠A 6= 90◦. Then, thethree primary trigonometric ratios for ∠A are:

Sine: sinA = oppositehypotenuse

Cosine: cosA = adjacenthypotenuse

Tangent: tanA = oppositeadjacent

The phrase SOH-CAH-TOA is a mnemonic for these ratios.

J. Garvin — Primary Trigonometric Ratios

Slide 5/10

tr i gonometry

Primary Trigonometric Ratios

There are six possible ratios of sides that can be made fromthe three sides.

The three primary trigonometric ratios are sine, cosine andtangent.

Primary Trigonometric Ratios

Let ∆ABC be a right triangle with ∠A 6= 90◦. Then, thethree primary trigonometric ratios for ∠A are:

Sine: sinA = oppositehypotenuse

Cosine: cosA = adjacenthypotenuse

Tangent: tanA = oppositeadjacent

The phrase SOH-CAH-TOA is a mnemonic for these ratios.

J. Garvin — Primary Trigonometric Ratios

Slide 5/10

tr i gonometry

Primary Trigonometric Ratios

There are six possible ratios of sides that can be made fromthe three sides.

The three primary trigonometric ratios are sine, cosine andtangent.

Primary Trigonometric Ratios

Let ∆ABC be a right triangle with ∠A 6= 90◦. Then, thethree primary trigonometric ratios for ∠A are:

Sine: sinA = oppositehypotenuse

Cosine: cosA = adjacenthypotenuse

Tangent: tanA = oppositeadjacent

The phrase SOH-CAH-TOA is a mnemonic for these ratios.

J. Garvin — Primary Trigonometric Ratios

Slide 5/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

sinA =opp

hyp

= 35

cosA =adj

hyp

= 45

tanA =opp

adj

= 34

J. Garvin — Primary Trigonometric Ratios

Slide 6/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

sinA =opp

hyp

= 35

cosA =adj

hyp

= 45

tanA =opp

adj

= 34

J. Garvin — Primary Trigonometric Ratios

Slide 6/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

sinA =opp

hyp

= 35

cosA =adj

hyp

= 45

tanA =opp

adj

= 34

J. Garvin — Primary Trigonometric Ratios

Slide 6/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

sinA =opp

hyp

= 35

cosA =adj

hyp

= 45

tanA =opp

adj

= 34

J. Garvin — Primary Trigonometric Ratios

Slide 6/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC . Express all ratios in simplest form.

sinA =opp

hyp

= 513

cosA =adj

hyp

= 1213

tanA =opp

adj

= 512

J. Garvin — Primary Trigonometric Ratios

Slide 7/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC . Express all ratios in simplest form.

sinA =opp

hyp

= 513

cosA =adj

hyp

= 1213

tanA =opp

adj

= 512

J. Garvin — Primary Trigonometric Ratios

Slide 7/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC . Express all ratios in simplest form.

sinA =opp

hyp

= 513

cosA =adj

hyp

= 1213

tanA =opp

adj

= 512

J. Garvin — Primary Trigonometric Ratios

Slide 7/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC . Express all ratios in simplest form.

sinA =opp

hyp

= 513

cosA =adj

hyp

= 1213

tanA =opp

adj

= 512

J. Garvin — Primary Trigonometric Ratios

Slide 7/10

tr i gonometry

Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

Use the PythagoreanTheorem to determinethe length of thehypotenuse, h.

h2 = 62 + 32

h2 = 45

h =√

45

J. Garvin — Primary Trigonometric Ratios

Slide 8/10

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Primary Trigonometric Ratios

Example

State the three primary trigonometric ratios for ∠A in∆ABC .

Use the PythagoreanTheorem to determinethe length of thehypotenuse, h.

h2 = 62 + 32

h2 = 45

h =√

45

J. Garvin — Primary Trigonometric Ratios

Slide 8/10

tr i gonometry

Primary Trigonometric Ratios

This gives us the following right triangle.

sinA =opp

hyp

= 3√45

cosA =adj

hyp

= 6√45

tanA =opp

adj

= 36

= 12

J. Garvin — Primary Trigonometric Ratios

Slide 9/10

tr i gonometry

Primary Trigonometric Ratios

This gives us the following right triangle.

sinA =opp

hyp

= 3√45

cosA =adj

hyp

= 6√45

tanA =opp

adj

= 36

= 12

J. Garvin — Primary Trigonometric Ratios

Slide 9/10

tr i gonometry

Primary Trigonometric Ratios

This gives us the following right triangle.

sinA =opp

hyp

= 3√45

cosA =adj

hyp

= 6√45

tanA =opp

adj

= 36

= 12

J. Garvin — Primary Trigonometric Ratios

Slide 9/10

tr i gonometry

Primary Trigonometric Ratios

This gives us the following right triangle.

sinA =opp

hyp

= 3√45

cosA =adj

hyp

= 6√45

tanA =opp

adj

= 36

= 12

J. Garvin — Primary Trigonometric Ratios

Slide 9/10

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Questions?

J. Garvin — Primary Trigonometric Ratios

Slide 10/10