Trigonometric Ratios , math homework help

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8-4 Trigonometry Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan, respectively. sin R = leg opposite ∠𝑅 hypotenuse cos R = 𝑟 leg adjacent to ∠𝑅 hypotenuse 𝑠 =𝑡 =𝑡 tan R = leg opposite ∠𝑅 leg adjacent to ∠𝑅 𝑟 =𝑠 Example: Find sin A, cos A, and tan A. Express each ratio as a fraction and a decimal to the nearest hundredth. sin A = opposite leg hypotenuse 𝐵𝐶 = 𝐵𝐴 = 5 13 ≈ 0.38 cos A = adjacent leg hypotenuse 𝐴𝐶 = 𝐴𝐵 = 12 13 ≈ 0.92 tan A = opposite leg adjacent leg 𝐵𝐶 = 𝐴𝐶 = 5 12 ≈ 0.42 Exercises Find sin J, cos J, tan J, sin L, cos L, and tan L. Express each ratio as a fraction and as a decimal to the nearest hundredth if necessary. 1. 2. 3. 8-4 (continued) Trigonometry Use Inverse Trigonometric Ratios You can use a calculator and the sine, cosine, or tangent to find the measure of the angle, called the inverse of the trigonometric ratio. Example: Use a calculator to find the measure of ∠T to the nearest tenth. The measures given are those of the leg opposite ∠T and the hypotenuse, so write an equation using the sine ratio. opp sin T = hyp 29 29 sin T = 34 29 If sin T = 34, then sin−1 34 = m∠T. Use a calculator. So, m∠T ≈ 58.5. Exercises Use a calculator to find the measure of ∠ T to the nearest tenth. 1. 2. 3. 4. 5. 6. Find sin L, cos L, tan L, sin M, cos M, and tan M. Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. ℓ = 15, m = 36, n = 39 2. ℓ = 12, m = 12√3, n = 24 Find x. Round to the nearest hundredth. 3. 4. 5. Use a calculator to find the measure of ∠B to the nearest tenth. 6. 7. 8. 8-4 Word Problem Practice Trigonometry 1. RADIO TOWERS Kay is standing near a 200-foot-high radio tower. Use the information in the figure to determine how far Kay is from the top of the tower. Express your answer as a trigonometric function. 2. RAMPS A 60-foot ramp rises from the first floor to the second floor of a parking garage. The ramp makes a 15° angle with the ground. How high above the first floor is the second floor? Express your answer as a trigonometric function. 3. TRIGONOMETRY Melinda and Walter were both solving the same trigonometry problem. However, after they finished their computations, Melinda said the answer was 52 sin 27° and Walter said the answer was 52 cos 63°. Could they both be correct? Explain. NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 4. LINES Jasmine draws line m on a coordinate plane. What angle does m make with the x-axis? Round your answer to the nearest degree. Angles of Elevation and Depression Name the angle of depression or angle of elevation in each figure 1. 2. 3. WATER TOWERS A student can see a water tower from the closest point of the soccer field at San Lobos High School. The edge of the soccer field is about 110 feet from the water tower and the water tower stands at a height of 32.5 feet. What is the angle of elevation if the eye level of the student viewing the tower from the edge of the soccer field is 6 feet above the ground? Round to the nearest tenth. 4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladder reaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of the ladder to the roof is 55°, how far is the ladder from the base of the wall? Round your answer to the nearest foot. 5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determine whether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from the flagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth? Chapter 8 25
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8-4 Trigonometry
Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a
trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which
are abbreviated sin, cos, and tan, respectively.
sin R =

leg opposite ∠𝑅
hypotenuse

cos R =

𝑟

leg adjacent to ∠𝑅
hypotenuse
𝑠

=𝑡

=𝑡

tan R =

leg opposite ∠𝑅
leg adjacent to ∠𝑅
𝑟

=𝑠

Example: Find sin A, cos A, and tan A. Express each ratio as a fraction and a
decimal to the nearest hundredth.
sin A =

opposite leg
hypotenuse
𝐵𝐶

= 𝐵𝐴
=

5
13

≈ 0.38

cos A =

adjacent leg
hypotenuse
𝐴𝐶

= 𝐴𝐵
=

12
13

≈ 0.92

tan A =

opposite leg
adjacent leg
𝐵𝐶

= 𝐴𝐶
=

5
12

≈ 0.42

Exercises
Find sin J, cos J, tan J, sin L, cos L, and tan L. Express each ratio as a fraction and as a decimal to the nearest
hundredth if necessary.
1.

2.

3.

Answers:
1.
𝟏𝟐
= 𝟎. 𝟔
𝟐𝟎
𝟏𝟔
𝒄𝒐𝒔𝑱 =
= 𝟎. 𝟖
𝟐𝟎
𝟏𝟐
𝒕𝒂𝒏𝑱 =
= 𝟎. 𝟕𝟓
𝟏𝟔
𝒔𝒊𝒏𝑱 =

𝟏𝟔
= 𝟎. 𝟖
𝟐𝟎
𝟏𝟐
𝒄𝒐𝒔𝑳 =
= 𝟎. 𝟔
𝟐𝟎
𝟏𝟔
𝒕𝒂𝒏𝑳 =
= 𝟏. 𝟑𝟑
𝟏𝟐
𝒔𝒊𝒏𝑳 =

2.
𝟐𝟒
= 𝟎. 𝟔
𝟒𝟎
𝟑𝟐
𝒄𝒐𝒔𝑱 =
= 𝟎...