The University of Toledo - Department of Mathematics and ...



Pre-Class Problems 9 for Wednesday, February 27

These are the type of problems that you will be working on in class. These problems are from Lesson 7.

Solution to Problems on the Pre-Exam.

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. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

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 [pic], x, and y. Then approximate the value of x and y to the nearest hundredth.

x

[pic]

y

34.7

[pic]

b. Find the exact value of [pic], x, and z. Then approximate the value of x and z to the nearest tenth.

[pic]

z

x

[pic]

24.2

Additional problems available in the textbook: Page 494 … 51 – 54.

Solutions:

1 a. [pic]

Answer: 0.5736

NOTE: In order to find the sine of the angle [pic], 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 [pic]. Since you know that the terminal side of the angle [pic] is in the second quadrant, where sine is positive, then you would know that this value is not correct.

Back to Problem 1.

1 b. [pic]

Answer: [pic]

NOTE: The secondary key of COS[pic], 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 [pic]. Do not round this number, which is [pic]. Now, find the multiplicative inverse (reciprocal) of this number using your reciprocal key, which is [pic] or [pic] , in order to obtain the secant of the angle [pic] since secant is the reciprocal of cosine.

NOTE: In order to find the cosine of the angle [pic], 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 [pic]. Since you know that the terminal side of the angle [pic] 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[pic], then your calculator would give you an error message since the number [pic] is greater than one. We will learn in Lesson 9 that you cannot take the inverse cosine of numbers greater than one.

Back to Problem 1.

1c. [pic]

Answer: [pic]

NOTE: The secondary key of TAN[pic], 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 [pic]. Do not round this number, which is [pic]. Now, find the multiplicative inverse (reciprocal) of this number using your reciprocal key, which is [pic] or [pic] , in order to obtain the cotangent of the angle [pic] since cotangent is the reciprocal of tangent.

NOTE: In order to find the tangent of the angle [pic], 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 [pic] for [pic]. If the mode of your calculator was set on Degrees and you used the secondary key of TAN[pic], then you would incorrectly give an answer of 89.8017 [pic]. Since you know that the terminal side of the angle [pic] is in the fourth quadrant, where cotangent is negative, then you would know that this value is not correct.

Back to Problem 1.

1d. [pic]

Answer: 0.4154

NOTE: In order to find the cosine of the angle [pic], 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. [pic]

Answer: [pic]

NOTE: In order to find the tangent of the angle [pic], 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 [pic].

Back to Problem 1.

1f. [pic]

Answer: 1.0095

NOTE: The secondary key of SIN[pic], 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 [pic] or [pic] , 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 [pic]. If the mode of your calculator was set on Radians and you used the secondary key of SIN[pic], 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 cannot take the inverse sine of numbers greater than one.

Back to Problem 1.

2a. x

[pic]

y

34.7

[pic]

To find [pic]: [pic]

Answer: [pic]

To find x: [pic]

Answer: Exact: [pic]

Approximate: 15.65

NOTE: [pic]

To find y: [pic]

Answer: Exact: [pic]

Approximate: 30.97

NOTE: [pic]

Back to Problem 2.

2b.

[pic]

z

x

[pic]

24.2

To find [pic]: [pic]

Answer: [pic]

To find x:

[pic]

OR [pic]

Answer: Exact: [pic] OR [pic]

Approximate: 20.8

NOTE: [pic]

[pic]

To find z:

[pic]

OR [pic]

Answer: Exact: [pic] OR [pic]

Approximate: 31.9

NOTE: [pic]

[pic]

Back to Problem 2.

Solution to Problems on the Pre-Exam: Back to Page 1.

13. Given the triangle below, find x. Set up an equation and solve. (4 pts.)

45

x

[pic]

[pic]

OR [pic]

Answer: [pic] OR [pic]

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