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Similar Triangles
MPM2D: Principles of Mathematics
In the diagram below, ∆ABC ∼ ∆ADE since ∠A is common
to both triangles, and ∠ACB = ∠AED.
Primary Trigonometric Ratios
J. Garvin
This means that any ratio of two sides in ∆ABC is equal to
the ratio of corresponding sides in ∆ADE .
J. Garvin — Primary Trigonometric Ratios
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Similar Triangles
Primary Trigonometric Ratios
By varying the measure of ∠A, the ratio of two sides in
∆ABC will change, but will remain equal to the ratio of
corresponding sides in ∆ADE .
In the right triangle ∆ABC below, the three sides have been
labelled based on their position relative to ∠A.
Therefore, a specific measure of ∠A can be associated with a
specific 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 also
increases. Thus, the ratio will increase.
If ∠A decreases, the length of the opposite side also
decreases. 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
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The opposite and adjacent sides are reversed relative to ∠B,
but the hypotenuse is always across from the right angle.
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Primary Trigonometric Ratios
Primary Trigonometric Ratios
There are six possible ratios of sides that can be made from
the three sides.
Example
The three primary trigonometric ratios are sine, cosine and
tangent.
State the three primary trigonometric ratios for ∠A in
∆ABC .
sin A =
Primary Trigonometric Ratios
Let ∆ABC be a right triangle with ∠A 6= 90◦ . Then, the
three primary trigonometric ratios for ∠A are:
opposite
Sine: sin A =
hypotenuse
adjacent
Cosine: cos A =
hypotenuse
opposite
Tangent: tan A =
adjacent
=
cos A =
=
tan A =
=
The phrase SOH-CAH-TOA is a mnemonic for these ratios.
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opp
hyp
3
5
adj
hyp
4
5
opp
adj
3
4
trigonometry
trigonometry
Primary Trigonometric Ratios
Primary Trigonometric Ratios
Example
Example
State the three primary trigonometric ratios for ∠A in
∆ABC . Express all ratios in simplest form.
State the three primary trigonometric ratios for ∠A in
∆ABC .
Use the Pythagorean
Theorem to determine
the length of the
hypotenuse, h.
sin A =
=
cos A =
=
tan A =
=
opp
hyp
5
13
adj
hyp
h2 = 62 + 32
h2 = 45
√
h = 45
12
13
opp
adj
5
12
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Primary Trigonometric Ratios
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Questions?
This gives us the following right triangle.
sin A =
opp
hyp
cos A =
adj
hyp
=
√3
45
=
√6
45
J. Garvin — Primary Trigonometric Ratios
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tan A =
=
=
opp
adj
3
6
1
2
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