"Applications of Trigonometric Functions" Trigonometric Functions PPT

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"Applications of Trigonometric Functions" Trigonometric Functions PPT

Part One: Explanation of Curriculum Standards

1. Understand that trigonometric functions are important function models that describe periodic change phenomena.

2. Be able to use trigonometric function models to solve simple practical problems.

3. Preliminarily learn to use analytical data or image features to perform some simple function fitting.

Application of trigonometric functions PPT, part 2: independent preview

1. Application of trigonometric functions

1. Simple harmonic motion

(1) For the function y=Asin(ωx+φ), which parameters are related to its maximum value and period? If it is a simple harmonic oscillation, its analytical formula is y=3sin(πx+π/6), combined with physics knowledge , what are its amplitude, period and initial phase respectively?

Tip: The maximum value of the function y=Asin(ωx+φ) is related to A, and the period is related to ω; for simple harmonic vibration y=3sin(πx+π/6), its amplitude is equal to 3, the period is 2, and the initial phase is π/6.

(2)Fill in the blanks

In y=Asin(ωx+φ),x∈[0,+∞)(A>0,ω>0), the physical meaning of each parameter.

(3) Do it

The amplitude, period and initial phase of simple harmonic oscillation y=1/3sin(x/4+π/5) are ()

A.1/3,π/2,π/5 B.1/3,8π,π/5

C.-1/3,8π,π/5 D.1/3,8π,4π/5

Analysis: Because A=1/3, ω=1/4, the period T=2π/(1/4)=8π, so the amplitude is 1/3, and the initial phase φ=π/5.

Answer:B

2. General procedures for applying trigonometric function models to solve problems

1. Fill in the blanks

To apply the trigonometric function model to solve problems, we must first abstract the actual problem into a mathematical problem, analyze its changing trend and determine its period, thereby establishing an appropriate trigonometric function model. The general procedure for solving the problem is as follows:

(1) Review the question, first review the question conditions, requirements, and understand the mathematical relationships.

(2) Model, analyze the periodicity of the question, and select an appropriate trigonometric function model.

(3) Solve and analyze the established trigonometric function model to obtain mathematical conclusions.

(4) Reduction, reducing mathematical conclusions to solutions to practical problems.

2. Do it

As shown in the figure, the water surface height h (unit: meter) of a bay relative to the mean sea level changes from 0 to 24 hours on a certain day. The functional relationship between the water surface height h and time t is ______________.

Application of trigonometric function models in daily life

Example 1 When the heart beats, blood pressure increases or decreases. The maximum and minimum values ​​of blood pressure are called systolic blood pressure and diastolic blood pressure respectively. The readings on the sphygmomanometer are the systolic blood pressure and diastolic blood pressure. The reading 120/80 mmHg is the standard value. Suppose Someone's blood pressure satisfies the functional formula p(t)=115+25sin 160πt, where p(t) is blood pressure (unit: mmHg), t is time (unit: min), try to answer the following questions:

(1) Find the period of function p(t);

(2) Find the number of heartbeats of this person per minute;

(3) Draw a sketch of function p(t);

(4) Find the reading of the person's blood pressure on the sphygmomanometer.

Analysis: The analytical formula of the function is known, and (1) (2) can be solved according to the period formula and the relationship between period and frequency. Use the "five-point drawing method" to solve (3). The function of the function can be obtained from the analytical formula or image of the function. The maximum and minimum values ​​are the blood pressure readings on the sphygmomanometer, thus obtaining (4).

Application of trigonometric functions PPT, the third part: thinking analysis

Failure to correctly understand the process of simple harmonic motion leads to errors

Typical example: A spring oscillator takes point O as the equilibrium position and performs simple harmonic motion between B and C. B and C are 20 cm apart. At a certain moment, the oscillator is at point B. After 0.5 s, the oscillator reaches point C for the first time. Find:

(1) Amplitude, period and frequency of vibration;

(2) The distance traveled by the oscillator in 5 s and the magnitude of the displacement at this time.

Wrong solution (1) Because B and C are 20 cm apart,

So the amplitude A=20 cm.

Because the oscillator reaches point C for the first time after 0.5 s from point B,

So the period T=0.5 s, the frequency f=1/T=2.

(2) The distance within 5 s = displacement = 5A = 5 × 20 = 100 cm.

Where is the misinterpretation? Can you spot it? How to avoid such mistakes?

Tip: The oscillator takes O as the equilibrium position and performs simple harmonic motion between B and C. The distance between B and C is 20 cm. This means that the maximum and minimum points of the oscillator leaving the equilibrium position are 20 cm apart, that is, twice the amplitude is equal to 20 cm. ;The oscillator first reaches point C from point B in 0.5 seconds, and then returns to point B. It is a cycle. Therefore, it should be =0.5 s. There is a difference between distance and displacement. The distance only has size, while displacement not only has size, but also direction. Wrong In the solution, due to unclear understanding of the concept of period, the period calculation is wrong. In addition, the direct difference between distance and displacement is confused, which leads to wrong results.

Application of trigonometric functions PPT, Part 4: Class drills

1. The phase and initial phase of the function y=3sin("-" x+π/6) are () respectively

A.-x+π/6,π/6 B.x-π/6,-π/6

C.x+5π/6,5π/6 D.x+5π/6,π/6

Analysis: Because y=3sin("-" x+π/6)

=3sin[π"-" ("-" x+π/6)]=3sin(x+5π/6),

So the phase and initial phase are x+5π/6, 5π/6 respectively.

Answer:C

2. The figure shows the position of each point on the rope at a certain moment for a rope wave propagating to the right. After the period, the position of B will move to ()

A. On the x-axis B. The lowest point C. The highest point D. Uncertain

Analysis: There is half a period between the adjacent maximum and minimum values, so B moves to the highest point.

Answer:C

3. As shown in the figure is an image of simple harmonic motion, then the following judgment is correct ()

A. The vibration period of the particle is 0.7 s

B. The amplitude of the particle is -5 cm

C. The particle has a maximum vibration speed at 0.1 s and 0.5 s.

D. The displacement of the particle is zero at 0.3 s and 0.7 s.

Analysis: According to the image in the question and the relevant knowledge of simple harmonic motion, T=0.8 s, A=5 cm. When t=0.1 s or 0.5 s, v is zero.

Answer:D

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"Applications of Trigonometric Functions" Trigonometric Functions PPT Download Part One: Learning Objectives 1. Understand that trigonometric functions are important function models that describe periodic changing phenomena, and be able to use trigonometric function models to solve some simple practical problems. (Key points) 2. Practical questions...

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Update Time: 2024-08-16

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