"End of Chapter Review Improvement Course" Trigonometric Functions PPT

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"End of Chapter Review Improvement Course" Trigonometric Functions PPT

Comprehensive improvement

Basic relations and induced formulas of same-angle trigonometric functions

It is known that cos(π+α)=-12, and the angle α is in the fourth quadrant, calculate:

(1)sin(2π-α);

(2) sin＋α＋（2n＋1）π＋sin（π＋α）sin（π－α）cos（α＋2nπ）(n∈Z)．

regular method

(1) Application of basic relations of trigonometric functions with same angles

① If one trigonometric function is known, find the other two: use square relationships and quotient relationships to solve directly or solve equations (sets) to solve.

② Given the tangent, find the homogeneous expression containing sine and cosine;

(i) When the homogeneous formula is a fraction, the numerator and denominator are both divided by cos α or cos2α, and then converted into a tangent.

(ii) When the homogeneous expression is an integer, the denominator is regarded as 1, and 1=sin2α+cos2α is substituted, and then the numerator and denominator are divided by cos α or cos2α.

(2) Simplify the evaluation method using induced formulas

① For the simplified evaluation of trigonometric function expressions, the key is to convert the angle into 2kπ±α, π±α, π2±α, 32π±α (or k·π2±α, k∈Z) according to the characteristics of the given angle. The form is simplified by "change from odd to even and remain unchanged, look at the symbols for quadrants".

②To solve the problem of "given the value of a certain trigonometric function, find the value of other trigonometric functions", the key is to observe and analyze the condition angle and conclusion angle, clarify the difference between the conditions and conclusions, connect the known and the unknown, and also Pay attention to the application of holistic thinking.

Graphics and Transformations of Trigonometric Functions

It is known that a lowest point on the graph of function f(x)=Asin(ωx+φ)(A>0,ω>0,0<φ<π2) is M2π3,-2, and the period is π.

(1) Find the analytical formula of f(x);

(2) Stretch the abscissa coordinates of all points on the image of y=f(x) to twice the original value (the ordinate remains unchanged), and then translate the resulting image π6 units to the right along the x-axis , get the graph of function y=g(x), and write the analytical formula of function y=g(x).

regular method

(1) Determine the parameters in the analytical formula y=Asin(ωx+φ) from the image or part of the image

①A: Determine A based on the maximum value and minimum value.

②ω: Determine ω by finding the period T.

③φ: Find it using the known point sequence equation.

(2) Two methods to transform the image of function y=sin x into the image of y=Asin(ωx+φ), (A>0, ω>0)x∈R

Properties of Trigonometric Functions

It is known that the function f(x)=4tan xsinπ2-x·cosx-π3-3.

(1) Find the domain and minimum positive period of f(x);

(2) Discuss the monotonicity of f(x) on the interval -π4, π4.

regular method

(1) Two properties of trigonometric functions

① Periodicity: The minimum positive period of the functions y=Asin(ωx+φ) and y=Acos(ωx+φ) is 2π|ω|, and the minimum positive period of y=tan(ωx+φ) is π|ω| ;.

② Parity: Odd functions in trigonometric functions can generally be reduced to y=Asin ωx or y=

Atan ωx, and the even function can generally be transformed into the form of y=Acos ωx+B.

(2) Method to find the value range (maximum value) of trigonometric functions

①Use the boundedness of sin x and cos x.

② Step by step analyze the range of ωx+φ from the form of y=Asin(ωx+φ)+k, and write the value range of the function based on the monotonicity of the sine function.

③Substitution method: Treating sin x or cos x as a whole can be reduced to the problem of finding the value range (maximum value) of the function in the interval.

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Update Time: 2024-09-07

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