Tangent Ratio (TOA)

Show your work for each problem.

1.
What is the tangent ratio formula?
1. a)
2. b)
3. c)
4. d)
Answer:
The tangent ratio is . The mnemonic TOA helps: Tangent = Opposite / Adjacent.
2.
What is the value of ?
1. a)
2. b)
3. c)
4. d)
Answer:
At , the opposite and adjacent sides are equal. Since .
3.
Which sides of a right triangle does tangent NOT use?
1. a)Hypotenuse
2. b)Opposite
3. c)Adjacent
4. d)All of the above
Answer: Hypotenuse
Tangent = Opposite / Adjacent. The hypotenuse is not used in the tangent ratio. This makes tangent unique among the three main trig ratios.
4.
What is ?
Answer: 0
At , the opposite side has length . So .
5.
In a right triangle with opposite side and adjacent side , what is ?
Answer: 1
. When opposite equals adjacent, the tangent equals , and the angle is .
6.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
In a 30-60-90 triangle, opposite to is and adjacent is . So .
7.
If , what is the simplified value?
Answer: 4/3
. The tangent value is .
8.
If the opposite side is and the adjacent side is , what is ? Give your answer as a decimal.
Answer: 0.75
.
9.
What happens to as approaches ?
1. a)It approaches 1
2. b)It stays constant
3. c)It approaches 0
4. d)It approaches infinity
Answer: It approaches infinity
As , the adjacent side approaches while the opposite stays positive. Dividing by an ever-smaller number makes tangent grow without bound toward infinity.
10.
A tree casts a shadow meters long. If the angle of elevation from the tip of the shadow to the top of the tree is , find the height of the tree. (Use )
Answer: 10.08
1. What is the adjacent side (shadow length)? 12
2. Write the tangent formula: 12
3. Solve for height: height = 10.08
11.
A ramp rises feet over a horizontal distance of feet. Find the angle of the ramp. (Use )
Answer: 14
1. What is the opposite side (rise)? 5
2. What is the adjacent side (run)? 20
3. Calculate 0.25
4. Find the angle: 14
12.
From a point meters from a building, the angle of elevation to the top is . Find the height of the building to the nearest meter. (Use )
Answer: 114
Height = meters.
13.
From the top of a cliff meters high, a boat is spotted at an angle of depression of . How far is the boat from the base of the cliff? (Use )
Answer: 319
1. What is the opposite side (cliff height)? 150
2. Write the formula: . Rearrange to find distance. 150/tan(25)
3. Calculate: 319
14.
A ladder reaches meters up a wall. The base of the ladder is meters from the wall. Find the angle the ladder makes with the ground. (Round to the nearest degree)
Answer: 66
1. What is the opposite side (height on wall)? 9
2. What is the adjacent side (distance from wall)? 4
3. Calculate 2.25
4. Find 66
15.
A surveyor stands meters from a tower. The angle of elevation to the top is . The surveyor's eyes are meters above the ground. Find the total height of the tower. (Use )
Answer: 64
1. Calculate the height above eye level: 62.5
2. Add the eye height to get total tower height: 64
16.
What is ? Express your answer in simplified radical form.
Answer: 1/√3
1. In a 30-60-90 triangle, what is the side opposite 30 degrees? 1
2. What is the side adjacent to 30 degrees (not the hypotenuse)? sqrt(3)
3. Calculate . This equals? 1/sqrt(3)