Chapter 2 Acute Angles and Right Triangles.

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1 Chapter 2 Acute Angles and Right Triangles

2 2.1 Trigonometric Functions of Acute Angles

3 Development of Right Triangle Definitions of Trigonometric Functions
Let ABC represent a right triangle with right angle at C and angles A and B as acute angles, with side “a” opposite A, side “b” opposite B and side “c” (hypotenuse) opposite C. Place this triangle with either of the acute angles in standard position (in this example “A”): Notice that (b,a) is a point on the terminal side of A at a distance “c” from the origin

4 Development of Right Triangle Definitions of Trigonometric Functions
Based on this diagram, each of the six trigonometric functions for angle A would be defined:

5 Right Triangle Definitions of Trigonometric Functions
The same ratios could have been obtained without placing an acute angle in standard position by making the following definitions: Standard “Right Triangle Definitions” of Trigonometric Functions ( MEMORIZE THESE!!!!!! )

6 Example: Finding Trig Functions of Acute Angles
Find the values of sin A, cos A, and tan A in the right triangle shown. A C B 52 48 20

7 Development of Cofunction Identities
Given any right triangle, ABC, how does the measure of B compare with A? B =

8 Cofunction Identities
By similar reasoning other cofunction identities can be verified: For any acute angle A, sin A = cos(90  A) csc A = sec(90  A) tan A = cot(90  A) cos A = sin(90  A) sec A = csc(90  A) cot A = tan(90  A)

9 Example: Write Functions in Terms of Cofunctions
b) sec 78 = csc (90  78) = csc 12 Write each function in terms of its cofunction. a) cos 38 = sin (90  38) = sin 52

10 Solving Trigonometric Equations Using Cofunction Identities
Given a trigonometric equation that contains two trigonometric functions that are cofunctions, it may help to find solutions for unknowns by using a cofunction identity to convert to an equation containing only one trigonometric function as shown in the following example

11 Example: Solving Equations
Assuming that all angles are acute angles, find one solution for the equation:

12 Comparing the relative values of trigonometric functions
Sometimes it may be useful to determine the relative value between trigonometric functions of angles without knowing the exact value of either one To do so, it often helps to draw a simple diagram of two right triangles each having the same hypotenuse and then to compare side ratios

13 Example: Comparing Function Values
Tell whether the statement is true or false. sin 31 > sin 29 Generalizing, in the interval from 0 to 90, as the angle increases, so does the sine of the angle Similar diagrams and comparisons can be done for the other trig functions

14 Equilateral Triangles
Triangles that have three equal side lengths are equilateral Equilateral triangles also have three equal angles each measuring 60o All equilateral triangles are similar (corresponding sides are proportional)

15 Using 30-60-90 Triangle to Find Exact Trigonometric Function Values
Find each of these:

16 Isosceles Right Triangles
Right triangles that have two legs of equal length Also have two angles of measure 45o All such triangles are similar

17 Using 45-45-90 Triangle to Find Exact Trigonometric Function Values
Find each of these:

18 Function Values of Special Angles
2 60 1 45 30 csc  sec  cot  tan  cos  sin  

19 Usefulness of Knowing Trigonometric Functions of Special Anlges: 30o, 45o, 60o
The trigonometric function values derived from knowing the side ratios of the and triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator You will often be asked to find exact trig function values for angles other than 30o, 45o and 60o angles that are somehow related to trig function values of these angles

20 Homework 2.1 Page 51 All: 1 – 14, 16 – 21, 23 – 26, 29 – 32, 35 – 42
MyMathLab Assignment 2.1 for practice MyMathLab Homework Quiz 2.1 will be due for a grade on the date of our next class meeting

21 2.2 Trigonometric Functions of Non-Acute Angles

22 Reference Angles A reference angle for an angle  is the positive acute angle made by the terminal side of angle  and the x-axis. (Shown below in red)

23 Example: Find the reference angle for each angle.
218 Positive acute angle made by the terminal side of the angle and the x-axis is: 218  180 = 38 1387 First find coterminal angle between 0o and 360o Divide 1387 by 360 to get a quotient of about 3.9. Begin by subtracting 360 three times. 1387 – 3(360) = 307 The reference angle for 307 is: 360 – 307  = 53

24 Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles
Each angle below has the same reference angle Choosing the same “r” for a point on the terminal side of each (each circle same radius), you will notice from similar triangles that all “x” and “y” values are the same except for sign

25 Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles
Based on the observations on the previous slide: Trigonometric functions of any angle will be the same value as trigonometric functions of its reference angle, except for the sign of the answer The sign of the answer can be determined by quadrant of the angle Also, we previously learned that the trigonometric functions of coterminal angles always have equal values

26 Finding Trigonometric Function Values for Any Non-Acute Angle 
Step 1 If  > 360, or if  < 0, then find a coterminal angle by adding or subtracting  as many times as needed to get an angle greater than 0 but less than 360. Step 2 Find the reference angle '. Step 3 Find the trigonometric function values for reference angle '. Step 4 Determine the correct signs for the values found in Step 3. (Hint: All students take calculus.) This gives the values of the trigonometric functions for angle .

27 Example: Finding Exact Trigonometric Function Values of a Non-Acute Angle
Find the exact values of the trigonometric functions for 210. (No Calculator!) Reference angle: 210 – 180 = 30 Remember side ratios for triangle. Corresponding sides:

28 Example Continued Trig functions of any angle are equal to trig functions of its reference angle except that sign is determined from quadrant of angle 210o is in quadrant III where only tangent and cotangent are positive Based on these observations, the six trig functions of 210o are:

29 Example: Finding Trig Function Values Using Reference Angles
Find the exact value of: cos (240) Coterminal angle between 0 and 360: 240 + 360 = 120 the reference angles is: 180  120 = 60

30 Expressions Containing Powers of Trigonometric Functions
An expression such as: Has the meaning: Example: Using your memory regarding side ratios of and triangles, simplify:

31 Example: Evaluating an Expression with Function Values of Special Angles
Evaluate cos 120 + 2 sin2 60  tan2 30. Individual trig function values before evaluating are: Substituting into the expression: cos 120 + 2 sin2 60  tan2 30

32 Finding Unknown Special Angles that Have a Specific Trigonometric Function Value
Example: Find all values of in the interval given: Use your knowledge of trigonometric function values of 30o, 45o and 60o angles* to find a reference angle that has the same absolute value as the specified function value Use your knowledge of signs of trigonometric functions in various quadrants to find angles that have both the same absolute value and sign as the specified function value *NOTE: Later we will learn to use calculators to solve equations that don’t necessarily have these special angles as reference angles

33 Example: Finding Angle Measures Given an Interval and a Function Value
Find all values of in the interval given: Which special angle has the same absolute value cosine as this angle? In which quadrants is cosine negative? Putting 45o reference angles in quadrants II and III, gives which two angles as answers?

34 Homework 2.2 Page 59 All: 1 – 6, 10 – 17, 25 – 32, 36 – 37, 48 – 53, MyMathLab Assignment 2.2 for practice MyMathLab Homework Quiz 2.2 will be due for a grade on the date of our next class meeting

35 2.3 Finding Trigonometric Function Values Using a Calculator

36 Function Values Using a Calculator
As previously mentioned, calculators are capable of finding trigonometric function values. When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. Also, angles measured in degrees, minutes and seconds must be converted to decimal degrees Remember that most calculator values of trigonometric functions are approximations.

37 Function Values Using a Calculator
Sine, Cosine and Tangent of a specific angle may be found directly on the calculator by using the key labeled with that function Cosecant, Secant and Cotangent of a specific angle may be found by first finding the corresponding reciprocal function value of the angle and then using the reciprocal key label x-1 or 1/x to get the desired function value Example: To find sec A, find cos A, then use the reciprocal key to find: This is the sec A value

38 Example: Finding Function Values with a Calculator
Convert 38 to decimal degrees and use sin key. Find tan of the angle and use reciprocal key cot 

39 Finding Angle Measures When a Trigonometric Function of Angle is Known
When a trigonometric ratio is known, and the angle is unknown, inverse function keys on a calculator can be used to find an angle* that has that trigonometric ratio Scientific calculators have three inverse functions each having an “apparent exponent” of -1 written above the function name. This use of the superscript -1 DOES NOT MEAN RECIPROCAL If x is an appropriate number, then gives the measure of an angle* whose sine, cosine, or tangent is x. * There are an infinite number of other angles, coterminal and other, that have the same trigonometric value

40 Example: Using Inverse Trigonometric Functions to Find Angles
Use a calculator to find an angle in the interval that satisfies each condition. Using the degree mode and the inverse sine function, we find that an angle having sine value is We write the result as

41 Example: Using Inverse Trigonometric Functions to Find Angles continued
Find one value of given: Use reciprocal identities to get: Now find using the inverse cosine function. The result is:

42 Homework 2.3 Page 64 All: 5 – 29, 55 – 62 MyMathLab Assignment 2.3 for practice MyMathLab Homework Quiz 2.3 will be due for a grade on the date of our next class meeting

43 2.4 Solving Right Triangles

44 Measurements Associated with Applications of Trigonometric Functions
In practical applications of trigonometry, many of the numbers that are used are obtained from measurements Such measurements many be obtained to varying degrees of accuracy The manner in which a measured number is expressed should indicate the accuracy This is accomplished by means of “significant digits”

45 Significant Digits “Digits obtained from actual measurement”
All digits used to express a number are considered “significant” (an indication of accuracy) if the “number” includes a decimal The number of significant digits in is: The number of significant digits in is: When a decimal point is not included, then trailing zeros are not “significant” The number of significant digits in 32,000 is: The number of significant digits in 50,700 is:

46 Significant Digits for Angles
The following conventions are used in expressing accuracy of measurement (significant digits) in angle measurements Tenth of a minute, or nearest thousandth of a degree 5 Minute, or nearest hundredth of a degree 4 Ten minutes, or nearest tenth of a degree 3 Degree 2 Angle Measure to Nearest: Number of Significant Digits

47 Calculations Involving Significant Digits
An answer is no more accurate than the least accurate number in the calculation Examples:

48 Solving a Right Triangle
To “solve” a right triangle is to find the measures of all the sides and angles of the triangle A right triangle can be solved if either of the following is true: One side and one acute angle are known Any two sides are known

49 Example: Solving a Right Triangle, Given an Angle and a Side
Solve right triangle ABC, if A = 42 30' and c = 18.4. How would you find angle B? B = 90  42 30' B = 47 30‘ = 47.5 A C B c = 18.4 4230'

50 Example: Solving a Right Triangle Given Two Sides
Solve right triangle ABC if a = cm and c = cm. A C B c = 27.82 a = 11.47

51 Angles of “Elevation” and “Depression”
Angle of Elevation: from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint X. Angle of Depression: from point X to point Y (below) is the acute angle formed by ray XY and a horizontal ray with endpoint X.

52 Solving an Applied Trigonometry Problem
Step 1 Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. Step 2 Use the sketch to write an equation relating the given quantities to the variable. Step 3 Solve the equation, and check that your answer makes sense.

53 Example: Application Shelly McCarthy stands 123 ft from the base of a flagpole, and the angle of elevation to the top of the pole is 26o40’. If her eyes are 5.30 ft above the ground, find the height of the pole.

54 Example: Application The length of the shadow of a tree m tall is m. Find the angle of elevation of the sun. Draw a sketch. The angle of elevation of the sun is 37.85. 22.02 m 28.34 m B

55 Homework 2.4 Page 72 All: 11 – 14, 21 – 28, 35 – 36, 41 – 44, 48 – 49
MyMathLab Assignment 2.4 for practice MyMathLab Homework Quiz 2.4 will be due for a grade on the date of our next class meeting

56 2.5 Further Applications of Right Triangles

57 Describing Direction by Bearing (First Method)
Many applications of trigonometry involve “direction” from one point to another Directions may be described in terms of “bearing” and there are two widely used methods The first method designates north as being 0o and all other directions are described in terms of clockwise rotation from north (in this context the angle is considered “positive”, so east would be bearing 90o)

58 Describing Bearing Using First Method
Note: All directions can be described as an angle in the interval: [ 0o, 360 ) Show bearings: 32o, 164o, 229o and 304o

59 Hints on Solving Problems Using Bearing
Draw a fairly accurate figure showing the situation described in the problem Look at the figure to see if there is a triangular relationship involving the unknown and a trigonometric function Write an equation and solve the problem

60 Example Radar stations A and B are on an east-west line 3.7 km apart. Station A detects a plane at C on a bearing of 61o, while station B simultaneously detects the same plane on a bearing of 331o. Find the distance from A to C.

61 Describing Direction by Bearing (Second Method)
The second method of defining bearing is to indicate degrees of rotation east or west of a north line or east or west of a south line Example: N 30o W would represent 30o rotation to the west of a north line Example: S 45o E would represent 45o rotation to the east of a south line

62 Example: Using Bearing
An airplane leaves the airport flying at a bearing of N 32 W for 200 miles and lands. How far west of its starting point is the plane? The airplane is approximately 106 miles west of its starting point. e 200

63 Using Trigonometry to Measure a Distance
A method that surveyors use to determine a small distance d between two points P and Q is called the subtense bar method. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. Angle  is measured, then the distance d can be determined.

64 Example: Using Trigonometry to Measure a Distance
Find d when  = and b = cm Let b = 2, change  to decimal degrees.

65 Example: Solving a Problem Involving Angles of Elevation
Sean wants to know the height of a Ferris wheel. He doesn’t know his distance from the base of the wheel, but, from a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3o . He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4o. Find the height of the Ferris wheel.

66 Example: Solving a Problem Involving Angles of Elevation continued
The figure shows two unknowns: x and h. Use the two triangles, to write two trig function equations involving the two unknowns: In triangle ABC, In triangle BCD, x C B h D A 75 ft

67 Example: Solving a Problem Involving Angles of Elevation continued
Since each expression equals h, the expressions must be equal to each other. Resulting Equation Distributive Property Get x-terms on one side. Factor out x. Divide by the coefficient of x.

68 Example: Solving a Problem Involving Angles of Elevation continued
We saw above that Substituting for x. tan 42.3 = and tan = So, tan tan = = and The height of the Ferris wheel is approximately 74 ft.

69 Homework 2.5 Page 81 All: 11 – 16, 23 – 28 MyMathLab Assignment 2.5 for practice MyMathLab Homework Quiz 2.5 will be due for a grade on the date of our next class meeting