Pre-Class Problems8 for Thursday, February21
Problems which are due at the beginning of class:
1.Use your calculator to approximate the following to four decimal places. (Round to the nearest ten-thousandth.)
a. b.
2.Find the exact value of , x, and z. Then approximate the value of x and z to the nearest tenth.
z
x
48.3
3.A 25-foot ladder is leaning against the top of a vertical wall. If the top of the ladder makes an angle of with the wall, then find the height of the wall. Find the exact value and then round to the nearest tenth.
4.One website that you used for these pre-class problems other than mine.
These are the type of problems that you will be working on in class. These problems are fromLesson 7.
You can go to the solution for each problem by clicking on the problem letter.
Objective of the following problems: To use a calculator to approximate the value of a trigonometric function of an angle.
1.Use your calculator to approximate the following to four decimal places. (Round to the nearest ten-thousandth.)
a. b. c.
d. e. f.
Objective of the following problems: To solve for unknowns in a given right triangle. To use a calculator to obtain approximations for the exact answers.
2.Solve for the following variables.
a.Find the exact value of , x, and y. Then approximate the value of x and y to the nearest hundredth.
x
y
34.7
b.Find the exact value of , x, and z. Then approximate the value of x and z to the nearest tenth.
z
x
24.2
Objective of the following problems: To take a written description and produce a right triangle with known information and one unknown. The unknown information is represented by a variable. Then use a trigonometric function to obtain an equation containing the variable. Solve this equation for the exact value of the variable. Then approximate the exact value of the variable as indicated.
For some of these problems, you will need the definition for angle of elevation and for angle of depression.
An angle of elevation and an angle of depression are both acute angles measured with respect to the horizontal. An angle of elevation is measured upward and an angle of depression is measured downward. The angle below is an angle of elevation from the point A to the point B above. The angle below is an angle of depression from the point B to the point A below.
B B
A A
3a.The angle of depression from the top of a building to an object on the ground is . If the object is 85 feet from the base of the building, then find the height of the building. Find the exact value and then round to the nearest tenth.
3b.The angle of depression from the top of a 150-foot building to an object on the ground is . How far is the object from the base of the building? Find the exact value and then round to the nearest hundredth.
3c.From a point P on the ground, the angle of elevation to the top of a 60-yard tree is . What is the distance from the point P to the top of the tree? Find the exact value and then round to the nearest hundredth.
3d.The angle of elevation of the string from the ground to a kite is . If the length of the string is 125 meters, then how far is the kite above ground? Find the exact value and then round to the nearest tenth.
3e.An observer on the ground is 105 yards from the point directly beneath a balloon. If the angle of elevation from the observer to the balloon is , then how far is the balloon from the observer? Find the exact value and then round to the nearest hundredth.
3f.A ladder is leaning against the top of a vertical wall. The top of the ladder makes an angle of with the wall. If the height of the wall is 6 meters, then find the length of the ladder. Find the exact value and then round to the nearest tenth.
3g.The angle of elevation from an object on the ground to the top of a building is . If the object is 95 meters from the top of the building, then find the distance from the object to the base of the building. Find the exact value and then round to the nearest thousandth.
3h.From a point on the ground which is 40 feet from the base of a tree, the angle of elevation to the top of the tree is . What is the height of the tree? Find the exact value and then round to the nearest tenth.
3i.A ladder is leaning against the top of a 15-yard vertical wall. The bottom of the ladder makes an angle of with the ground. How far is the bottom of the ladder from the base of the wall? Find the exact value and then round to the nearest hundredth.
Additional problems available in the textbook: Evaluating Trigonometric Functions with a Calculator - Parts a and b on Page 157. Examples 7 and 9 on page 158. Page 161 … 63, 64, 65, 66, 67, 70, 71. Page 211 … 5, 6, 7, 8, 19, 20, 21, 22. Examples 1 and 2 on Page 205.
Requires a system of equations to solve: Page 162 … 72. Page 211 … 23, 24, 25. For Problem 23, use for . Examples 3 on Page 206.
Solutions:
1a.
Answer: 0.5736
NOTE: In order to find the sine of the angle , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of . Since you know that the terminal side of the angle is in the second quadrant, where sine is positive, then you would know that this value is not correct.
Back to Problem 1.
1b.
Answer:
NOTE: The secondary key of COS, which is above the COS key, on your calculator is NOT the secant key. It is the key for the inverse cosine function which we will study in Lesson 9.
NOTE: Since your calculator does not have a secant key, you will first need to find the cosine of the angle . Do not round this number, which is . Now, find the multiplicative inverse (reciprocal) of this number using your reciprocal key, which is or , in order to obtain the secant of the angle since secant is the reciprocal of cosine.
NOTE: In order to find the cosine of the angle , the mode of your calculator needs to be set on Radians. If your calculator is set on Degrees, then you would incorrectly give an answer of 1.0025 for . Since you know that the terminal side of the angle is in the third quadrant, where secant is negative, then you would know that this value is not correct. If the mode of your calculator was set on Radians and you used the secondary key of COS, then your calculator would give you an error message since the number is greater than one. We will learn in Lesson 9 that you can not take the inverse cosine of numbers greater than one.
Back to Problem 1.
1c.
Answer:
NOTE: The secondary key of TAN, which is above the TAN key, on your calculator is NOT the cotangent key. It is the key for the inverse tangent function which we will study in Lesson 9.
NOTE: Since your calculator does not have a cotangent key, you will first need to find the tangent of the angle . Do not round this number, which is . Now, find the multiplicative inverse (reciprocal) of this number using your reciprocal key, which is or , in order to obtain the cotangent of the angle since cotangent is the reciprocal of tangent.
NOTE: In order to find the tangent of the angle , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of for . If the mode of your calculator was set on Degrees and you used the secondary key of TAN, then you would incorrectly give an answer of 89.8017. Since you know that the terminal side of the angle is in the fourth quadrant, where cotangent is negative, then you would know that this value is not correct.
Back to Problem 1.
1d.
Answer: 0.4154
NOTE: In order to find the cosine of the angle , the mode of your calculator needs to be set on Radians. If your calculator is set on Degrees, then you would incorrectly give an answer of 0.9916.
Back to Problem 1.
1e.
Answer:
NOTE: In order to find the tangent of the angle , the mode of your calculator needs to be set on Degrees. If your calculator is set on Radians, then you would incorrectly give an answer of .
Back to Problem 1.
1f.
Answer: 1.0095
NOTE: The secondary key of SIN, which is above the SIN key, on your calculator is NOT the cosecant key. It is the key for the inverse sine function which we will study in Lesson 9.
NOTE: Since your calculator does not have a cosecant key, you will first need to find the sine of the angle 14 (radians). Do not round this number, which is 0.9906073557. Now, find the multiplicative inverse (reciprocal) of this number using your reciprocal key, which is or , in order to obtain the cosecant of the angle 14 (radians) since cosecant is the reciprocal of sine.
NOTE: In order to find the sine of the angle 14 (radians), the mode of your calculator needs to be set on Radians. If your calculator is set on Degrees, then you would incorrectly give an answer of 4.1336 for . If the mode of your calculator was set on Radians and you used the secondary key of SIN, then your calculator would give you an error message since the number 14 is greater than one. We will learn in Lesson 9 that you can not take the inverse sine of numbers greater than one.
Back to Problem 1.
2a. x
y
34.7
To find :
Answer:
To find x:
Answer:Exact:
Approximate: 15.65
NOTE:
To find y:
Answer:Exact:
Approximate: 30.97
NOTE:
Back to Problem 2.
2b.
z
x
24.2
To find :
Answer:
To find x:
OR
Answer:Exact: OR
Approximate: 20.8
NOTE:
To find z:
OR
Answer:Exact: OR
Approximate: 31.9
NOTE:
Back to Problem 2.
3a. Top of Building ------
y
Object
85 feet
NOTE: Since the angle of depression is , then the angle of elevation is also .
Answer:Exact: ft
Approximate: 71.3 ft
Back to Problem 3.
3b. Top of Building ------
150 feet
Object
x
NOTE: Since the angle of depression is , then the angle of elevation is also .
OR
Answer:Exact: ft OR ft
Approximate: 326.12 ft
Back to Problem 3.
3c. Top of Tree
z
60 yards
P
OR
Answer:Exact: yd OR yd
Approximate: 104.61 yd
Back to Problem 3.
3d. Kite
125 meters
y
Answer:Exact: m
Approximate: 93.8 m
Back to Problem 3.
3e. Balloon
z
Observer
105 yards
OR
Answer:Exact: yd OR yd
Approximate: 118.92 yd
Back to Problem 3.
3f. Top of Wall
z
6 meters
OR
Answer:Exact: m OR m
Approximate: 7.2 m
Back to Problem 3.
3g. Top of Building
95 meters
Object
x
Answer:Exact: m
Approximate: 51.741 m
Back to Problem 3.
3h. Top of Tree
y
40 feet
Answer:Exact: ft
Approximate: 125.3 ft
Back to Problem 3.
3i. Top of Wall
15 yards
x
OR
Answer:Exact: yd OR yd
Approximate: 33.53 m
Back to Problem 3.
Solution to Problems on the Pre-Exam:
13.Given the triangle below, find x. Set up an equation and solve. (4 pts.)
45
x
OR
Answer: OR
15.From the top of a building, which is 80 meters tall, the angle of depression to an object on level ground below is . How far is the object from the top of the building? Draw a picture and label known information. Indicate any variable you use. Set up an equation and solve. (6 pts.)
Top of Building ------
z
80 meters
Object
NOTE: Since the angle of depression is , then the angle of elevation is also .
OR
Answer: m OR m