Lesson 98 Using Sum And Difference Formulas
Essential Question How can you evaluate trigonometric functions of the sum or difference of two angles?
EXPLORATION 1
Work with a partner.a. Explain why the two triangles shown are congruent.b. Use the Distance Formula to write an expression for d in the first unit circle.c. Use the Distance Formula to write an expression for d in the second unit circle.d. Write an equation that relates the expressions in parts and . Then simplify this equation to obtain a formula for cos.
EXPLORATION 2
Deriving a Sum FormulaWork with a partner. Use the difference formula you derived in Exploration 1 to write a formula for cos in terms of sine and cosine of a and b. Hint: Use the fact that cos = cos.
EXPLORATION 3
Deriving Difference and Sum FormulasWork with a partner. Use the formulas you derived in Explorations 1 and 2 to write formulas for sin and sin in terms of sine and cosine of a and b. Hint: Use the cofunction identities sin a)= cos a and cos a)= sin a and the fact thatcos = sin and sin = sin.
Communicate Your Answer
How can you evaluate trigonometric functions of the sum or difference of two angles?Answer:
Question 5.a. Find the exact values of sin 75° and cos 75° using sum formulas. Explain your reasoning.b. Find the exact values of sin 75° and cos 75° using difference formulas. Compare your answers to those in part .Answer:
Find the exact value of the expression.Question 1.
cos 15° = \
Explanation:= cos 60° cos 45° + sin 60° sin 45°= \ . \ + \ . \= \ + \= \
tan \
cos \ = \
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Lesson 96 Modeling With Trigonometric Functions
Essential Question What are the characteristics of the real-life problems that can be modeled by trigonometric functions?
EXPLORATION 1
Modeling Electric CurrentsWork with a partner. Find a sine function that models the electric current shown in each oscilloscope screen. State the amplitude and period of the graph.
Communicate Your Answer
What are the characteristics of the real-life problems that can be modeled by trigonometric functions?Answer:
Question 3.Use the Internet or some other reference to find examples of real-life situations that can be modeled by trigonometric functions.Answer:
Question 1.WHAT IF?In Example 1, how would the function change when the audiometer produced a pure tone with a frequency of 1000 hertz?
Answer:
Explanation:Lets suppose that the audiometer produced a pure tone with a frequency of 100 hz. The maximum pressure is 2. so a = 2f = \
Question 11.MODELING WITH MATHEMATICSThe lowest frequency of sounds that can be heard by humans is 20 hertz. The maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write and graph a sine model that gives the pressure P as a function of the time t.Answer:
Question 12.MODELING WITH MATHEMATICSA middle-A tuning fork vibrates with a frequency f of 440 hertz . You strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write and graph a sine model that gives the pressure Pas a function of the time t .
Answer:The sine model is P = 5 sin
Answer:
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Lesson 94 Graphing Sine And Cosine Functions
Essential Question What are the characteristics of the graphs of the sine and cosine functions?
EXPLORATION 1
Graphing the Sine FunctionWork with a partner.a. Complete the table for y= sin x, where x is an angle measure in radians.b. Plot the points from part . Draw a smooth curve through the points to sketch the graph of y = sin x.c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over 2 x 2. Is the sine function even, odd, or neither?
EXPLORATION 2
Graphing the Cosine FunctionWork with a partner.a. Complete a table for y= cos x using the same values of x as those used in Exploration 1.b. Plot the points from part and sketch the graph of y= cos x.c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over 2 x 2. Is the cosine function even, odd, or neither?
Communicate Your Answer
What are the characteristics of the graphs of the sine and cosine functions?Answer:
Describe the end behavior of the graph of y = sin xAnswer:
Monitoring Progress
Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function.Question 1.g = \sin x
Answer:
Using Trigonometric Identities 97 Exercises
Vocabulary and Core Concept CheckQuestion 1.Describe the difference between a trigonometric identity and a trigonometric equation.Answer:
Question 2.WRITINGExplain how to use trigonometric identities to determine whether sec = sec or sec = sec .
Answer:
sec = \cos = cossec = \ = \ = sec
Monitoring Progress and Modeling with Mathematics
In Exercises 310, find the values of the other five trigonometric functions of .Question 3.sin = \, 0 < < \Answer:
sin = \, < < \
Answer:cos = \tan = \cot = \sec = \csc = \
Explanation:sin = \cos² = 1 sin²= 1 )²= \cos = \tan = \ = \ = \cot = \ = \ = \sec = \ = \csc = \ = \
Question 5.tan = \, \ < < Answer:
cot = \, \ < <
Answer:sin = \cos = \tan = \sec = \csc = \
Explanation:cot = \\ = \cos = \sin cos² + sin² = 1\sin² + sin² = 1sin² = \sin = \cos = \ . \= \tan = \ = \ = \sec = \ = \csc = \ = \
Question 7.cos = \, < < \Answer:
sec = \, \ < < 2
Answer:sin = \cos = \tan = \cot = \csc = \
Explanation:sec = \\ = \cos = \cos² + sin² = 1sin² = 1 cos² = 1 )sin = \tan = \ = \ = \cot = \ = \ = \csc = \ = \
Question 9.cot = 3, \ < < 2Answer:
csc = \, < < \
Answer:sin = \cos = \tan = \cot = \sec = \
Explanation:csc = \\ = \sin = \cos² + sin² = 1cos² = 1 sin² = 1 )cos = \tan = \ = \ = \cot = \ = \ = \sec = \ = \
\Answer:
Answer:\ = sin² x
\} \)Answer:
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Big Ideas Math Book Algebra 2 Answer Key Chapter 9 Trigonometric Ratios And Functions
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0.5² = 0.3² +b²b² = 0.25 0.09
Question 11.ABSTRACT REASONINGThe line segments connecting the points , , and form a triangle. Is the triangle a right triangle? Justify your answer.
Answer:AB = BC = AC = ² + ²AC² = ² + ² = AB² + BC²So the points form a right triangle.
Trigonometric Ratios And Functions Mathematical Practices
Mathematically proficient students reason quantitatively by creating valid representations of problems.
Monitoring Progress
Find the exact coordinates of the point on the unit circle.Question 1.
Evaluate the six trigonometric functions of the angle .Question 1.
sin = \cos = \tan = \cot = \sec = \csc = \
Explanation:hypotenuse = 4² + 3² = 5opposite side = 3sin = \ = \cos = \ = \tan = \ = \cot = \ = \sec = \ = \csc = \ = \
Question 2.
sin = \cos = \tan = \cot = \sec = \csc = \
Explanation:
sin = \ = \= \cos = \ = \= \tan = \ = \ = 1cot = \ = \ = 1sec = \ = \ = 2csc = \ = \ = 2
Question 4.In a right triangle, is an acute angle and cos = \. Evaluate the other five trigonometric functions of .
Answer:sin = \tan = \cot = \sec = \csc = \
Explanation:10² = 7² + x²x² = 100 49 = 51x = 51
BC² = AB² + AC²= 2² + 8² = 4 + 64BC = 217
Question 11.WHAT IF?In Example 6, estimate the height of the parasailer above the boat when the angle of elevation is 38°.
Answer:The height of the parasailer is 44.28 ft.
Explanation:sin 38° = \0.615 = \
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Trigonometric Ratios And Functions Study Skills: Form A Final Exam Study Group
9.19.4 What Did You Learn?
Core Vocabulary
Mathematical PracticesQuestion 1.Make a conjecture about the horizontal distances traveled in part of Exercise 39 on page 483.Answer:
Question 2.Explain why the quantities in part of Exercise 56 on page 493 make sense in the context of the situation.Answer:
Study Skills: Form a Final Exam Study Group
Form a study group several weeks before the final exam. The intent of this group is to review what you have already learned while continuing to learn new material.
Lesson 95 Graphing Other Trigonometric Functions
Essential Question What are the characteristics of the graph of the tangent function?
EXPLORATION 1
Graphing the Tangent FunctionWork with a partner. a. Complete the table for y = tan x, where x is an angle measure in radians.b. The graph of y = tan x has vertical asymptotes at x-values where tan x is undefined. Plot the points from part . Then use the asymptotes to sketch the graph of y = tan x.c. For the graph of y = tan x, identify the asymptotes, the x-intercepts, and the intervals for which the function is increasing or decreasing over \ x \. Is the tangent function even, odd, or neither?
Communicate Your Answer
What are the characteristics of the graph of the tangent function?Answer:
Question 3.Describe the asymptotes of the graph of y = cot x on the interval \ < x < \.Answer:
Monitoring Progress
Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.Question 1.
The function is in the form of g = atana = 1, b = 2Period = \ = \The x-intercepts are , 0) = \, 0)The halfway points are, a) = , 1), -a) = , -1)The vertical asymptotes are odd multiples of \x = \x = \The graph of g is a horizontal shrink by a factor of 0.5 of the graph of the function f = tan x
Question 2.g = \cot x
Answer:
Question 3.
Question 4.
Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.Question 5.
Question 15.Use the given graph to graph each function.Answer:
g = \ sec 2x
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Presentation On Theme: Algebra 2 Chapter 9 Conic Sections: Circles And Parabolas Presentation Transcript:
1 Algebra 2 Chapter 9 Conic Sections: Circles and Parabolas
2 9-2 Circles WARMUP: Give the constant that makes each of the following a perfect square trinomial: 1. x 2 + 8x + ____ 2. y 2 24x + ____ 3. x 2 x + ____ 4. y 2 + 3y + ____
3 9-2 Circles Objective: To learn the relationship between the center and radius of a circle and the equation of a circle.
4 9-2 Circles As we saw previously, the term Conics comes from slices of a double cone creating certain shapes. The simplest of these is the circle: Circle: the set of all points in a plane that are a fixed distance, the radius, from a fixed point, the center.
5 9-2 Circles C P Start with the distance formula: Every circle has an equation of this form.
6 9-2 Circles Equation of a circle: The circle with center and radius r has the equation:
7 9-2 Circles What is the equation of a circle with center and radius 5?
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Lesson 92 Angles And Radian Measure
Essential Question How can you find the measure of an angle in radians?Let the vertex of an angle be at the origin, with one side of the angle on the positive x-axis. The radian measure of the angle is a measure of the intercepted arc length on a circle of radius 1. To convert between degree and radian measure, use the fact that \ = 1.
EXPLORATION 1
Writing Radian Measures of AnglesWork with a partner. Write the radian measure of each angle with the given degree measure. Explain your reasoning.
EXPLORATION 2
Writing Degree Measures of AnglesWork with a partner. Write the degree measure of each angle with the given radian measure. Explain your reasoning.
Communicate Your Answer
How can you find the measure of an angle in radians?Answer:
Question 4.The figure shows an angle whose measure is 30 radians. What is the measure of the angle in degrees? How many times greater is 30 radians than 30 degrees? Justify your answers.Answer:
Draw an angle with the given measure in standard position.Question 1.
Explain how the sign of an angle measure determines its direction of rotation.
Answer:If the sign of an angles measure is positive, the direction of rotation of its terminal side will be counter clock wise from the positive x-axis.If the sign of angles measure is negative, the direction of rotation of its terminal side will be clockwise from the positive x-axis.
Question 3.In your own words, define a radian.Answer:
Monitoring Progress and Modeling with Mathematics
sin \
cot )
Trigonometric Ratios And Functions Chapter Review
9.1 Right Triangle Trigonometry
Question 1.In a right triangle, is an acute angle and cos = \. Evaluate the other five trigonometric functions of .
Answer:sin = \tan = \cot = \sec = \csc =\
Explanation:cos = \adj side = 6, hypotenuse = 11opposite side = 85sin = \ = \tan = \ = \cot = \ = \sec = \ = \csc = \ = \
Question 2.The shadow of a tree measures 25 feet from its base. The angle of elevation to the Sun is 31°. How tall is the tree?
Answer:The height of the tree is 15 ft
Explanation:tan 31° = \0.6 = \
sin = \ = \cos = = \ = \ = 0tan = \ = \ = undefinedcot = = \ = \ = 0sec = \ = \ = undefinedcsc = \ = \ = 1
Question 10.
sin = \cos = \tan = \cot = \sec =\csc = \
Explanation:sin = \ = \cos = = \ = \tan = \ = \cot = = \ = \sec = \ = \csc = \ = \
Question 11.
sin = \cos = \tan = \cot = \sec =\csc = \
Explanation:sin = \ = \cos = = \ = \tan = \ = \cot = = \ = \sec = \ = \csc = \ = \
Evaluate the function without using a calculator.Question 12.
sin \
Answer:sin \ = \
Explanation: = \ = \ 2 = \sin \ = sin\ = \
Question 15.sec \
Answer:sec \ = 2
Explanation: = \ = 4 \ = \sec \ = sec\ = 2
9.4 Graphing Sine and Cosine Functions
Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of the parent function.Question 16.
9.5 Graphing Other Trigonometric Functions
Answer:
Question 23.
Question 26.g = sec \x
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