Evaluating Sine, Cosine And Tangent
Evaluating Sine, Cosine and TangentI.Evaluate an Expressiona. To evaluate an expression means to a given value in fora variable andb. Evaluate the following:i. 3x if x 6ii. -4x2 -7x 2 if x -6II.Sine, Cosine and Tangenta. Sine, Cosine and Tangent are functions that arerelated to triangles and anglesi. We will discuss more about where they come from later! b. We can evaluate a , or just like anyother expressionc. We have buttons on our calculator for sine, cosine and tangenti. Sine ii. Cosine iii. Tangent d. When evaluating sine, cosine or tangent, we must remember that the value wesubstitute into the expression represents an .e. Angles are measured ini.ii.f. We have to check our mode to make sure the calculator knows what measure weare using!i. In this class, we will always use Degrees, but you should know that radiansexist! Make sure Degree is highlighted!g. Evaluate the following, round to the nearest thousandth:1. sin (52o)2. cos (122o)3. tan (-76o)4. cos (45o)5. sin (30o)6. tan (5 radians)2
Exploring Sine, Cosine and Tangent Angle RestrictionsUsing your calculator, complete the 01501802102402703003303601. What do you notice about the sine column? Describe the pattern.2. What do you notice about the cosine column? Describe the pattern.3. What do you notice about the tangent column? Describe the pattern.3
Evaluating Trigonometric FunctionsNameDateEvaluate each of the following using your calculator (round to the nearest thousandth.1. sin (62o)2. cos (132o)3. tan (-87o)4. cos (178o)5. sin (-60o)6. sin (78o)7. cos (-13o)8. tan (95o)9. cos (778o)10. sin (225o)11. tan (90 o)12. sin (3.4 radians)4
Solving Sine, Cosine and Tangent EquationsI.Solving Equationsa. To solve an equation means to “ ” all the operations to get thevariable by itselfb. To “undo” an operation means to use thei.The inverse operation of addition isii.The inverse operation of multiplication isiii.The inverse operation of squaring isc. Solve the following equations using inverse operations:II.i.3x 5 14ii.2x2 4 76Solving Sine, Cosine and Tangent Equationsa. We can solve equations involving , andjust like any other equation!b. Inverse operations of sine, cosine and tangenti.Sine ii.Cosine iii.Tangent c. Solve the following equations and express your answer in degrees:1. sin (x) 0.62. cos (x) 1.53. tan (x) -6.74. cos (x) -0.875. sin (x) 0.55
Solving Sine, Cosine and Tangent EquationsNameDateSolve the following equations and express your answer in degrees:1. sin (x) 0.82. cos (x) -1.73. tan (x) -9.54. cos (x) -0.785. sin (x) 0.3666. sin (x) -0.7687. -1cos (x) -0.728. 3tan (x) -12.89. 4cos (x) – 6 -5.210. 3sin (x) 4 1.5711. tan (x) 3.2712. 2sin (x) 5sin (x) – 6 -26
Pythagorean Theorem and SOHCAHTOA (find missing sides)I.Review: Pythagorean Theorema. Pythagorean Theorem is used to find missing sides in a triangle.b. “a” and “b” represent thec. “c” represents thed. Examples: Find the missing sides using Pythagorean Theoremi.2.3.II.4.SOHCAHTOAa. SOHCAHTOA is used to help find missing sides and angles in a right triangle whenPythagorean Theorem does not work!S (sine)O (opposite)H (hypotenuse) C (cosine)A (adjacent)H (hypotenuse) T (tangent) O (opposite)A (adjacent) 7
b. Setting up Trigonometry Ratios and Solving for Sidesi.(NOT the right angle)ii.(Opposite, Adjacent, Hypotenuse)iii.: if we have the opposite and hypotenuse if we have the adjacent and the hypotenuse if we have the opposite and the adjacentiv.Set up the proportion and solve for x!Example:1. Select a given angle2. Label your sides3. Decide which Trig to use4. Set up the proportion5. Solve the proportion6. Check your work!8
9
10
SOHCAHTOA (find missing angles)I.Review: SOHCAHTOASOHII.CAHTOASetting up Trigonometry Ratios and Solving for Anglesi. Select a given angle (NOT the right angle)ii. Label your sides (Opposite, Adjacent, Hypotenuse)iii. Decide which trig function you can use: SOH if we have the opposite and hypotenuse CAH if we have the adjacent and the hypotenuse TOA if we have the opposite and the adjacentiv. Solve the equation remember to you your inverses!Example:Find the measure of angle A.1. Select a given angle2. Label your sides3. Decide which Trig to use11
4. Set up the proportion5. Solve the equation6. Check your work!III.Find the measure of both missing angles:1.2.3.12
13
14
Angles of Elevation and DepressionI.Angles of Elevation and Depressiona. The angle of elevation is the angle formed by a and the lineof sight .b. The angle of depression is the angle formed by a and the lineof sight .c. Notice the angle of elevation and the angle of depression arewhen in the same picture!15
Angles of Elevation & DepressionFind all values to the nearest tenth.1. A man flies a kite with a 100 foot string. The angle of elevation of thestring is 52 o . How high off the ground is the kite?2. From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base ofthe cliff is 34º. How far is the object from the base of the cliff?3. An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must theplane take off in order to avoid crashing into the building? Assume that the airplane flies in astraight line and the angle of elevation remains constant until the airplaneflies over the building.4. A 14 foot ladder is used to scale a 13 foot wall. At what angle of elevation must the ladder be situated inorder to reach the top of the wall?5. A person stands at the window of a building so that his eyes are 12.6 m above the level ground. An object ison the ground 58.5 m away from the building on a line directly beneath the person. Compute the angle ofdepression of the person’s line of sight to the object on the ground.6. A ramp is needed to allow vehicles to climb a 2 foot wall. The angle of elevation in order for the vehicles tosafely go up must be 30 o or less, and the longest ramp available is 5 feetlong. Can this ramp be used safely?16
Area of a Triangle using SineI.Area of a Triangle using Sinea. Area of a triangle can be found using the following formula:b. Unfortunately, we are not always given the base and height!c. To find the height, we create a right triangle and use SOHCAHTOA!Example: Find the area of the triangle1. Drop the height to create a righttriangle2. Use SOHCAHTOA to solve forthe missing height3. Use the found height and thegiven base to calculate area17
18
19
Law of SinesI.Law of Sinesa. Law of Sines is used to find missing sides and angles in oblique trianglesb. Law of Sines can be used in the following cases:1. Angle-Angle-Side2. Angle-Side-Angle3. Side-Side-Angle (SPECIAL CASE!!)c. Examples: Find the missing sides and angles.1.2.20
Law of Sines (Ambiguous Case)I.Law of Sines-Ambiguous Casea. Ambiguous Case is used when you haveb. It is called ambiguous because we could have more than one answer!c. Solving an ambiguous case:1. Set up the problem and solve using2. If you get one solution,3. Find the second possible solutionFORMULA 4. Test the second angleTEST Example: Solve using Ambiguous Case1. Set up and solve using Law ofSines2. Find second possible solution3. Test second solution21
4. Complete problem withsolution(s)Example2: Solve Using Ambiguous Case1. Set up and solve using Law ofSines2. Find second possible solution3. Test second solution4. Complete problem withsolution(s)22
23
24
Law of CosinesI.Law of Cosinesa. Law of Cosines is used to find missing sides and angles in oblique trianglesb. Law of Cosines can be used in the following cases:1. Side-Side-Side2. Side-Angle-Sidec. Examples: Find the missing sides and angles:1.2.25
26
27
28
29
Amplitude and Midline (equation to graph)I.Amplitudea. A graph in the form or has anamplitude of .b. The amplitude of a standard or graph is.c. The amplitude of a sine or cosine graph can be found from an equation using thefollowing formula:d. Find the amplitude for each of the following:1. y 3sinx2. y -4cos5x3. y (1/3)sinx 5II.Midlinea. The midline is the line thatb. The midline is halfway between the andc. The midline can be found from an equation using the following formula:d. When there is no vertical shift, the midline is always .30
Exploring Amplitude and MidlineName1. Complete the following table:Degreey sinxY 3sinx03060901201501802102402703003303602. Graph y sinx3. Graph y 3sinx4. How are the graphs alike? How are they different?31
1. Complete the following table:Degreey sinxY (1/2)sinx03060901201501802102402703003303602. Graph y sinx3. Graph y (1/2)sinx4. How are the graphs alike? How are they different?32
Amplitude and Midline (graph to equation)III.Amplitude and Midlinea. The amplitude can be found from a graph by using the following formula:b. The midline can be found from a graph by using the following formula:c. Find the amplitude and midline for each of the following graphs:1.3.2.4.33
Exploring Sine, Cosine and Tangent GraphsName1. Complete the table below:DegreeSinxPoint (Degree, Sinx)03060901201501802102402703003303602. Using the points above (degree, sinx), sketch a graph of y sinx.34
3. Complete the table below:DegreeCosxPoint (Degree, Cosx)03060901201501802102402703003303604. Using the points above (degree, cosx), sketch a graph of y cosx.35
5. Complete the table below:DegreeTanxPoint (Degree, Tanx)03060901201501802102402703003303606. Using the points above (degree, tanx), sketch a graph of y tanx.7. What happens at 90o and 270o? Why?36
CCMII Unit 5 Part 2 Lesson 4 Graphing and Understanding Sine, Cosine and TangentI.II.Sine Grapha. Sine is increasing:c. Sine is positive:b. Sine is decreasing:d. Sine is negative:Cosine Grapha. Cosine is increasing:c. Cosine is positive:b. Cosine is decreasing:d. Cosine is negative:37
III.Tangent Grapha. Tangent is increasing:c. Tangent is positive:b. Tangent is decreasing:d. Tangent is negative:38
NameDateGraphing Sine and CosinePractice WorksheetGraph the following functions over two periods, one in the positive direction and one in the negativedirections. Label the axes appropriately.1. 𝑦 3cos (𝑥)Amplitude:Midline:2. 𝑦 4sin (x)Amplitude:Midline:39
3. 𝑦 2cos (𝑥)Amplitude:Midline:4. 𝑦 0.5sin (𝑥)Amplitude:Midline:5. 𝑦 5sin (𝑥) 1Amplitude:Midline:40
Sine and Cosine Graphs PracticeI.NameDateMatch each equation with the correct graph.A. 𝑦 𝑐𝑜𝑠 (𝑥)B. 𝑦 𝑠𝑖𝑛 (𝑥)C. 𝑦 𝑐𝑜𝑠 (𝑥) D. 𝑦 𝑠𝑖𝑛 (𝑥)E. 𝑦 𝑡𝑎𝑛(𝑥) F. 𝑦 𝑡𝑎𝑛(𝑥)41
II.Understanding Sine and Cosine in the calculator1. If you were graphing 𝑦 𝑠𝑖𝑛 (𝑥) in the calculator, what would your window need to be?Xmin: Xmax: Xscl: Ymin:Ymin: Yscl:2. If you were graphing 𝑦 𝑐𝑜𝑠 (𝑥) in the calculator, what would your window need to be?Xmin: Xmax: Xscl: Ymin:Ymin: Yscl:3. If you were graphing 𝑦 𝑡𝑎𝑛 (𝑥) in the calculator, what would your window need to be?Xmin: Xmax: Xscl: Ymin:Ymin: Yscl:4. Complete the following table using the graphs sitiveNegativePeriodMidlineABC42
a. Sine, Cosine and Tangent are _ functions that are related to triangles and angles i. We will discuss more about where they come from later! b. We can evaluate a _, _ or _ just like any other expression c. We have buttons on our calculator for sine, cosine and tangent i. Sine ii. Cosine iii. Tangent d.
Lesson 8-4 Sine and Cosine Ratios 439 Sine and Cosine Ratios The tangent ratio, as you have seen, involves both legs of a right triangle.The sine and cosine ratios involve one leg and the hypotenuse. of &A of &A These equations can be abbreviated: sin A cos A Writing Sine and Cosine Ratios Use the triangle to write each ratio. a. sin T .
Lesson 1: Graphs of the Sine, Cosine, and Tangent Objectives: Graph the sine, cosine, and tangent functions. State all values in the domain of a basic trigonometric function that correspond to a given value of the range. Graph transformations of the sine, cosine, and tangent graphs. Warm Up ! a. Use the graph of sin to state all values of for which sin is -1.
Section 5.2 - Graphs of the Sine and Cosine Functions In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations. Much of what we will do in graphing these problems wi
Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant The Cotangent Function The tangent function is cotx cosx sinx. It will have zeros where the cosine function has zeros, and vertical asymptotes where the sine function has zeros. It will look like the cosine function where the sine is essentially equal to 1, which is when xis near .
Explain why the cosine function is a periodic function. 4. Math Journal Draw the graphs for the sine function and the cosine function. Compare and contrast the two graphs. 5. Determine if the function is periodic. If so, state the period. Find each value by referring to the graph of the sine or the cosine function. 6. cos 2 7. sin 5 2 8.
the three ratios, sine, cosine, and tangent, defi ned in a right triangle 3.2 The Sine and Cosine Ratios MHR 117 003_M10CH03_FINAL.indd 1173_M10CH03_FINAL.indd 117 111/26/09 5:51:54 PM1/26/09 5:51:54 PM
(b) If the power of sine is odd , save one sine factor and use to express the remaining factors in terms of cosine: Then substitute . [Note that if the powers of both sine and cosine are odd, either (a) or (b) can be used.] (c) If the powers of both sine and cosine are even, use the half-ang
Artificial intelligence (AI) is a significant step forward in the digitalisation and transformation of modern businesses. In short, it refers to computers’ capability to acquire and apply knowledge without programmers’ intervention. Investors are lining up to be part of the imminent change. AI attracted USD 24 bn in investments globally in 2018, a twelvefold increase since 2013. US start .
