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tr i gonometry
MPM2D: Principles of Mathematics
Primary Trigonometric Ratios
J. Garvin
Slide 1/10
tr i gonometry
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
tr i gonometry
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
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
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
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
tr i gonometry
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
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
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
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
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
tr i gonometry
Questions?
J. Garvin — Primary Trigonometric Ratios
Slide 10/10