Cosine Ratio (CAH)

Show your work for each problem.

1.
In a right triangle, the cosine of an angle equals:
1. a)Hypotenuse / Adjacent
2. b)Opposite / Adjacent
3. c)Adjacent / Hypotenuse
4. d)Opposite / Hypotenuse
Answer: Adjacent / Hypotenuse
CAH stands for Cosine = Adjacent / Hypotenuse. The cosine of an angle is always the ratio of the adjacent side to the hypotenuse.
2.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This is a special angle value from the 30-60-90 triangle. Notice that because these are complementary angles.
3.
What is ?
Answer: 1
. This is the maximum value of cosine. At 0 degrees, the adjacent side equals the hypotenuse, giving us .
4.
In a right triangle, the adjacent side is and the hypotenuse is . What is ?
Answer: 0.8
.
5.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. In a 30-60-90 triangle, the side adjacent to the 30 degree angle is times the short leg. Also, .
6.
A cable is meters long and makes a angle with the ground. How far from the base is the cable anchored? (Use cos(50) = 0.643, round to 1 decimal place)
Answer: 7.7
Adjacent = hypotenuse x cos(50) = 12 x 0.643 = 7.716 meters, which rounds to 7.7 meters.
7.
A ramp covers a horizontal distance of meters and makes a angle with the ground. How long is the ramp? (Use cos(15) = 0.966)
Answer: 8.28
1. What is the adjacent side (horizontal distance)? 8
2. What is the unknown we're looking for? hypotenuse
3. Write the formula: hypotenuse = adjacent / cos(theta). What is 8 / 0.966? 8.28
8.
A helicopter travels km at a climb angle of . How much horizontal distance does it cover? (Use cos(25) = 0.906)
Answer: 2.72
1. What is the hypotenuse (flight path distance)? 3
2. What are we finding? (the horizontal distance) adjacent
3. Calculate: adjacent = 3 x 0.906 = ? 2.72
9.
Which formula would you use to find the hypotenuse when you know the adjacent side and the angle?
1. a)hypotenuse = cos(theta) / adjacent
2. b)hypotenuse = adjacent + cos(theta)
3. c)hypotenuse = adjacent x cos(theta)
4. d)hypotenuse = adjacent / cos(theta)
Answer: hypotenuse = adjacent / cos(theta)
Starting with cos(theta) = adjacent/hypotenuse, multiply both sides by hypotenuse: hypotenuse x cos(theta) = adjacent. Then divide both sides by cos(theta): hypotenuse = adjacent / cos(theta).
10.
A ladder is meters long and its base is meters from the wall. What angle does the ladder make with the ground? (Round to the nearest degree)
Answer: 71
1. Calculate cos(theta) = adjacent/hypotenuse = 2/6 = ? 0.333
2. To find the angle, use inverse cosine: theta = cos^-1(0.333). What is the result in degrees? 71
11.
A support wire for a tower is meters long and makes a angle with the ground. How far from the tower base is the wire anchored? (Use cos(55) = 0.574, round to 1 decimal place)
Answer: 14.4
Distance = 25 x cos(55) = 25 x 0.574 = 14.35 meters, which rounds to 14.4 meters.
12.
In a right triangle, the adjacent side is cm and the hypotenuse is cm. Find the angle theta. (Round to the nearest degree)
Answer: 53
1. Calculate cos(theta) = adjacent/hypotenuse = 9/15 = ? 0.6
2. Find theta = cos^-1(0.6). What is the angle in degrees? 53
13.
What is the value of ? (Give your answer as a decimal rounded to 3 decimal places)
Answer: 0.707
. This value comes from a 45-45-90 isosceles right triangle. Note that cos(45) = sin(45).
14.
If , what is the value of theta?
1. a)30 degrees
2. b)90 degrees
3. c)60 degrees
4. d)45 degrees
Answer: 60 degrees
. This is because . Remember: .
15.
A roof has a horizontal span of meters. If the roof makes a angle with the horizontal, how long is the slanted roof surface? (Use cos(20) = 0.940, round to 2 decimal places)
Answer: 12.77
1. What is the adjacent side (horizontal span)? 12
2. What is the unknown (slanted roof surface)? hypotenuse
3. Calculate hypotenuse = adjacent / cos(20) = 12 / 0.940 = ? 12.77