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Chapter 2

Chapter 2. Acute Angles and Right Triangles. 2.1. Trigonometric Functions of Acute Angles. Development of Right Triangle Definitions of Trigonometric Functions.

gabriel-bailey
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Chapter 2

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

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

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

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

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

  19. Usefulness of Knowing Trigonometric Functions of Special Anlges: 30o, 45o, 60o • The trigonometric function values derived from knowing the side ratios of the 30-60-90 and 45-45-90 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. 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 Example: Find the reference angle for each angle.

  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 byquadrant 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 360 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. Find the exact values of the trigonometric functions for 210. (No Calculator!) Reference angle: 210 – 180 = 30 Remember side ratios for 30-60-90 triangle. Corresponding sides: Example: Finding Exact Trigonometric Function Values of a Non-Acute Angle

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

  30. Expressions Containing Powers of Trigonometric Functions • An expression such as: • Has the meaning: • Example: Using your memory regarding side ratios of 30-60-90 and 45-45-90 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, 61- 66 • 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. Convert 38 to decimal degrees and use sin key. Find tan of the angle and use reciprocal key cot 68.4832  .3942492 Example: Finding Function Values with a Calculator

  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 .8535508 is 58.6 . 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: 66.289824

  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 583.104 is: • The number of significant digits in .0072 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. Number of Significant Digits Angle Measure to Nearest: 2 Degree 3 Ten minutes, or nearest tenth of a degree 4 Minute, or nearest hundredth of a degree 5 Tenth of a minute, or nearest thousandth of a degree Significant Digits for Angles • The following conventions are used in expressing accuracy of measurement (significant digits) in angle measurements

  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. B c = 18.4 4230' A C 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

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

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