5.3 Trigonometric Ratios

Trigonometric Ratios

1. Identify the reference angle

2. The Opposite Leg is across from the reference angle

3. The Adjacent Leg is next to the reference angle

4. The Hypotenuse is always across from the right angle

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Finding trig ratios

Multiple Choice

Find the Tangent of <c

• 27/36
• 27/45
• 45/36
• 45/27

Multiple Choice

Find the Tangent of <c

• 3/4

• 3/5

• 5/4

• 5/3

Multiple Choice

Find cos C

• 30 / 34
• 16 / 34
• 30 / 16
• 16 / 30

Multiple Choice

Find cos A

• 30 / 34
• 16 / 34
• 30 / 16
• 16 / 30

Multiple Choice

Given the reference angle shown which trig ratio would work for the problem.

• $$\tan\ x\ =\frac{400}{300}$$

• $$\sin\ x=\frac{300}{400}$$

• $$\tan\ x=\frac{300}{400}$$

• $$\cos\ x=\frac{300}{400}$$

Multiple Choice

Which trig ratio would be correct for the figure?

• $$\sin\left(?\right)=\frac{13}{18}$$

• $$\tan\left(?\right)=\frac{13}{18}$$

• $$\tan\left(?\right)=\frac{18}{13}$$

• $$\cos\left(?\right)=\frac{13}{18}$$

Consider the right triangle ABC above:
If ∠A=θ then, ∠C=90∘−θ

What are complementary angles in trigonometry?

​sinΘ = cos(90-Θ)
cosΘ = sin(90-Θ)

complementary angles are the set of two angles such that their sum is equal to 90°

Multiple Choice

Find ratio for cos X:

• ⁸/₁₅

• ¹⁷/₁₅

• ¹⁵/₁₇

• ⁸/₁₇

Multiple Choice

Now find the ratio for sin Z

• ⁸/₁₅

• ¹⁷/₁₅

• ¹⁵/₁₇

• ⁸/₁₇

Multiple Choice

If sin⁡30∘=1/2, then cos ? =1/2.

• 30

• 45

• 60

• 90

Remember: the sine of an angle is equal to the cosine of (90-angle)

The sine and cosine of complementary angles are equal.
Since (90∘−30∘)=60∘ is complementary to 30∘,
Therefore, cos⁡ 60∘= 1/2.

The blue triangle is dilated by a scale factor of 2 to form the green triangle. How does enlarging or reducing a triangle affect the trig ratios?

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​Trig Ratios of Dilated Triangles

Multiple Choice

Consider the pre-image triangle. Find $$\cos\theta$$

• $$\frac{3}{4}$$

• $$\frac{4}{3}$$

• $$\frac{4}{5}$$

• $$\frac{3}{5}$$

Multiple Choice

If a 3-4-5 right triangle is dilated by a scale factor of 2, what are the image sides?

• 2-2-2

• 1-2-3

• 5-6-7

• 6-8-10

Multiple Choice

The 6-8-10 triangle is a dilation of a 3-4-5 triangle. Find the ratio for $$\cos\theta$$

• $$\frac{6}{10}$$

• $$\frac{8}{10}$$

• $$\frac{6}{8}$$

• $$\frac{8}{6}$$

Multiple Choice

The 6-8-10 triangle is a dilation of a 3-4-5 triangle. Find the ratio for $$\cos\theta$$ REDUCE YOUR ANSWER

• $$\frac{6}{10}\Longrightarrow\frac{3}{5}$$

• $$\frac{8}{10}\Longrightarrow\frac{4}{5}$$

• $$\frac{6}{8}\Longrightarrow\frac{3}{4}$$

• $$\frac{8}{6}\Longrightarrow\frac{4}{3}$$

Remember similar triangles have the SAME corresponding angles.
Trig ratios of similar triangles will always reduce to be the same ratio.

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​Similar Triangles have the SAME trig ratios

Multiple Choice

Find $$\sin F$$

• not enough information

• $$\frac{45}{53}$$

• $$\frac{45}{28}$$

• $$\frac{53}{45}$$

​The ratios of their corresponding side lengths are equal and the measures of their corresponding angles are equal.

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These are similar by AA

Multiple Choice

Find $$\cos I$$

• $$\frac{12}{37}$$

• $$\frac{12}{35}$$

• $$\frac{35}{37}$$

• these are not similar triangles

Fill in the Blanks

Fill in the Blanks

Multiple Choice

What is the correct ratio?

• sin 40 = w/28
• cos 40 = w/28
• tan 40 = w/28
• sin 40 = 28/w

Multiple Choice

What is the correct trig ratio?

• $$\sin\left(28\degree\right)=\frac{x}{14}$$

• $$\tan\left(28\degree\right)=\frac{14}{x}$$

• $$\tan\left(28\degree\right)=\frac{x}{14}$$

• $$\cos\left(x\degree\right)=\frac{14}{28}$$