"Applications of Trigonometric Functions" Trigonometric Functions PPT Download

"Applications of Trigonometric Functions" Trigonometric Functions PPT Download

Part One: Learning Objectives

1. Understand that trigonometric functions are important function models that describe periodic change phenomena, and be able to use trigonometric function models to solve some simple practical problems. (emphasis)

2. The actual problem is abstracted into a trigonometric function model. (difficulty)

core competencies

1. Solve practical problems by establishing triangular models and cultivate mathematical modeling literacy.

2. Improve mathematical operation literacy with the help of practical problem solving.

Application of trigonometric functions PPT, part 2: independent preview and exploration of new knowledge

A preliminary exploration of new knowledge

1. The physical meaning of the parameters in function y=Asin(ωx+φ), A>0, ω>0

2. Basic steps for solving trigonometric function word problems:

(1) Review the meaning of the question;

(2) Collect and organize data and establish mathematical models;

(3) Discuss variable relationships and solve mathematical models;

(4) Test and draw conclusions.

First try

1. The period, amplitude, and initial phase of the function y=13sin13x+π6 are ()

A． 3π, 13, π6 B． 6π, 13, π6

C． 3π, 3, -π6 D. 6π, 3, π6

2. The frequency of the function y=3sin12x-π6 is ________, the phase is ________, and the initial phase is ________.

3. As shown in the figure is the image of a simple harmonic motion. This simple harmonic motion requires ________s round trip.

4. The image shown in the figure shows the change of the water surface height y (m) of a bay relative to the mean sea level within 24 hours on a certain day. The water surface height y is a function of the time x starting from 0 o'clock at night. The relational expression is ____________________.

PPT on the application of trigonometric functions, the third part: cooperative exploration to improve literacy

Application of trigonometric function model in physics

[Example 1] It is known that when a small ball hanging on a spring vibrates up and down, the displacement s (cm) of the small ball from the equilibrium position changes with time t (s) as s=4sin2t+π3, t∈[0 , +∞). Use the "five-point method" to make a simplified diagram of this function and answer the following questions.

(1) What is the displacement of the ball when it starts to vibrate (t=0)?

(2) What is the displacement of the ball when it rises to the highest point and when it drops to the lowest point?

(3) How long does it take for the ball to vibrate back and forth?

[Ideas Enlightenment] Determining the physical meanings of the parameters A, ω, and φ in the function y=Asin(ωx+φ) is the key to solving the problem.

Draw points and connect lines, and the image will appear as shown in the figure.

(1) Substituting t=0 into s=4sin2t+π3, we get s=4sin π3=23, so the displacement of the ball when it starts to vibrate is 23 cm.

(2) The displacement of the ball when it rises to the highest point and drops to the lowest point is 4 cm and -4 cm respectively.

(3) Because the period of vibration is π, the time it takes for the ball to vibrate back and forth once is π s.

regular method

In physics, when an object performs simple harmonic motion, the sinusoidal function y=Asinωx+φ can be used to express the change of the displacement y of the object's vibration with time x. A is the amplitude, which represents the maximum distance of the object from the equilibrium position, T=2πω is the period, which represents the time required for the object to vibrate back and forth once, and f=1T is the frequency, which represents the number of times the object vibrates back and forth per unit time.

Practical Applications of Trigonometric Function Models

[Inquiry Questions]

How many steps are typically required when dealing with curve fitting and prediction problems?

Tips: (1) Give a scatter plot based on the original data.

(2) By examining the scatter plot, draw the straight line or curve that is "closest" to it, that is, the fitting straight line or the fitting curve.

(3) Based on the learned function knowledge, find the functional relationship formula of the fitted straight line or fitted curve.

(4) Use functional relationship expressions to predict and control given problems according to conditions to provide a basis for decision-making and management.

Class summary

1. The application of curve y=Asin (ωx+φ) is essentially physical knowledge. Therefore, the establishment of mathematical models for this type of problem must be combined with physical knowledge.

2. The basic steps for solving trigonometric function application problems can be divided into four steps: problem review, modeling, model solving, and reduction evaluation.

(1) Construct a trigonometric function model to solve practical problems with periodic variation phenomena.

(2) Problems in measurement are reduced to triangles, and the concepts of trigonometric functions and the knowledge of solution triangles are used to solve the problems.

PPT on the application of trigonometric functions, part 4: Achieving standards in class and solidifying double bases

1. Thinking and analysis

(1) The period of the function y=|sin x+12| is π.()

(2) The period of a spring oscillator doing simple harmonic vibration is 0.4 s and the amplitude is 5 cm. Then the distance traveled by the oscillator in 2 s is 50 cm.()

(3) The relationship between current intensity I(A) and time t(s) is I=5sin100πt+π3. Then when t=1200 s, current intensity I is 52 A. ()

2. There are two small balls with masses M1 and M2 on each of the two springs that vibrate freely up and down. It is known that their displacements s1 (cm) and s2 (cm) from the equilibrium position at time t (s) are determined by s1=5sin2t+π6 and s2=10cos 2t respectively. Then when t=2π3 s, the relationship between s1 and s2 is ()

A． s1>s2 B． s1

C． s1=s2 D． Can not be sure

3. A line with a length of l cm is fixed at one end and hangs a small ball at the other end. The functional relationship between the displacement s (cm) of the ball from the equilibrium position when it swings and the time t (s) is s=3cosglt+π3, where g is the gravity Acceleration, when the ball swings with a period of 1 s, the line length is l = _________cm.

4. As shown in the figure, the number of a certain animal population is as low as 700 on January 1 and as high as 900 on July 1. Its total number changes according to a sinusoidal curve between these two values.

(1) Find the functional expression of population quantity y with respect to time t; (where t is measured in months since the beginning of the year)

(2) Estimated animal population size on March 1 of that year.

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"Applications of Trigonometric Functions" Trigonometric Functions PPT courseware:

"Applications of Trigonometric Functions" Trigonometric Functions PPT Courseware Part One Content: Learning Objectives Understand that trigonometric functions are important function models that describe periodic changing phenomena. Use trigonometric function models to solve simple practical problems... ... ... trigonometric functions Apply PPT, Chapter..