Trigonometry of Acute Angles

4.2.1: Trigonometry

What are two other right triangle ratios that are like tangent? How can I use them to solve a right triangle?

Review:

Find the missing side using the tangent ratio.

Review:

Find the missing side using the tangent ratio.

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#56

While traveling around the beautiful city of San Francisco, Juanisha climbs several steep streets. One of the steepest, Filbert Street, has a slope angle of  $17.5\degree$  , according to her guidebook.

Poll

Once Juanisha finishes walking 100 feet up the hill, she decides to figure out how high she has climbed. Juanisha draws the diagram below to represent this situation.

Can a tangent ratio be used to solve for Δy? Why or why not? Be prepared to share your thinking with the rest of the class.

• Yes

• No

Sine Ratio

There are two new ratios. One is the sine ratio. This is the ratio of the length of the side opposite the acute angle to the length of the hypotenuse. It is written:

 $\sin\left(60\degree\right)=\frac{\sqrt{3}}{2}$  

Cosine Ratio

The other is the cosine ratio. This is the ratio of the length of the side adjacent (next to) the acute angle to the length of the hypotenuse. It is written:

 $\cos\left(60\degree\right)=\frac{1}{2}$  

Multiple Choice

Which ratio should you use to find the missing side of this triangle?

• sine

• cosine

• tangent

Multiple Choice

Which equation would you use to solve for the missing side of the triangle?

• $$\sin\left(23\degree\right)=\frac{15}{a}$$

• $$\sin\left(23\degree\right)=\frac{a}{15}$$

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Multiple Choice

Which ratio should you use to find the missing side of this triangle?

• sine

• cosine

• tangent

Multiple Choice

Which equation would you use to solve for the missing side of the triangle?

• $$\cos\left(37\degree\right)=\frac{b}{8}$$

• $$\cos\left(37\degree\right)=\frac{8}{b}$$

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You Try

1. Label the sides of each triangle as "opposite", "adjacent", or "hypotenuse".

2. Decide which ratio to use based on the labeled sides.

3. Write an equation for each and solve for the missing side.

Multiple Choice

Which trigonometric ratio should you use?

• Tangent Ratio

• Sine Ratio

• Cosine Ratio

• Any ratio

Multiple Choice

Which trigonometric ratio should you use?

• Tangent Ratio

• Sine Ratio

• Cosine Ratio

• Any ratio

Multiple Choice

Which trigonometric ratio should you use?

• Tangent Ratio

• Sine Ratio

• Cosine Ratio

• Any ratio

Multiple Choice

Which trigonometric ratio should you use?

• Tangent Ratio

• Sine Ratio

• Cosine Ratio

• Any ratio

Multiple Choice

Find the missing value.

Round to the nearest tenths.

• 5.9 lightyears

• 5.0 lightyears

• 7.1 lightyears

• 7.5 lightyears

Multiple Choice

Find the missing value.

Round to the nearest tenths.

• 23.6 miles

• 4.6 miles

• .9 miles

• 20.7 miles

Multiple Choice

Find the missing value.

Round to the nearest tenths.

• 36.6 in

• 43.7 in

• 47.8 in

• 37.4 in

Multiple Choice

Find the missing value.

Round to the nearest tenths.

• 33.9 mm

• 3.4 mm

• 10.5 mm

• 36.2 mm