"End of Chapter Review Lesson" Trigonometric Functions PPT

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"End of Chapter Review Lesson" Trigonometric Functions PPT

Application of basic relations and induced formulas of same-angle trigonometric functions

[Example 1] (1) It is known that sin (-π + θ) + 2cos (3π - θ) = 0, then sin θ + cos θ sin θ - cos θ = ________.

(2) It is known that f(α)=sin2π-α·cos2π-α·tan-π＋α sin-π＋α·tan-α＋3π.

①Simplify f(α);

②If f(α)=18, and π4＜α＜π2, find the value of cos α-sin α;

③If α=-47π4, find the value of f(α).

[Ideas]First use the induction formula to simplify, and then use the basic relations of congruent trigonometric functions to evaluate.

Motif exploration

1. In this example (2), change "18" to "-18" and "π4＜α＜π2" to "-π4＜α＜0" to find cos α＋sin α.

[ Solution

So cos α＋sin α＞0,

And (cos α＋sin α)2＝1＋2sin αcos α＝1＋2×-18＝34, so cos α＋sin α＝32.

2. Let tan α in this example (2) be expressed as 1fα＋cos2α.

[Solution] 1fα＋cos2α＝1sin αcos α＋cos2α

＝sin2α＋cos2αsin αcos α＋cos2α＝tan2α＋1tan α＋1.

regular method

1. Keep in mind the two basic relations sin2α+cos2α=1 and sin αcos α=tan α, and be able to apply the two relations to evaluate, simplify and prove trigonometric functions. In application, pay attention to mastering problem-solving skills. For example: if the value of sin α±cos α is known, cos αsin α can be found. Note the application (cos α±sin α)2=1±2sin αcos α.

2. The induction formula can be summarized as the simplified formula of each trigonometric function value of k·π2±α (k∈Z). The memory rule is: the odd and even changes remain unchanged, and the symbols look at the quadrants.

Image transformation problem of trigonometric functions

[Example 2] (1) It is known that curve C1: y=cos x, C2: y=sin2x+2π3, then the following conclusion is correct ()

A． Stretch the abscissa of each point on C1 to twice its original value, and keep the ordinate unchanged. Then translate the obtained curve to the right by π6 unit lengths to obtain curve C2.

B. Stretch the abscissa of each point on C1 to twice its original value, and keep the ordinate unchanged. Then translate the obtained curve to the left by π12 unit lengths to obtain curve C2.

C． Shorten the abscissa coordinate of each point on C1 to 12 times its original value, keep the ordinate unchanged, and then translate the obtained curve to the right by π6 unit length to obtain curve C2

D. Shorten the abscissa coordinate of each point on C1 to 12 times its original value and keep the ordinate unchanged. Then translate the obtained curve to the left by π12 unit lengths to obtain curve C2.

(2) After translating the image of the function y=sin(2x+φ) to the left along the x-axis by π8 unit lengths, an image of an even function is obtained. Then one possible value of φ is ()

A.π2　B.π4

C． 0D. -π4

regular method

1. Two methods to transform the image of function y=sin x into the image of y=Asin(ωx+φ), x∈R

2. Symmetric transformation

(1) The image of y=f(x)→the image of y=-f(x) that is symmetrical about the x-axis.

(2) The image of y=f(x)→the image of y=f(-x) that is symmetrical about the y-axis.

(3) The image of y=f(x)→the image of y=-f(-x) that is symmetric about 0,0.

Properties of Trigonometric Functions

[Example 3] (1) If the function f(x)=3sin(2x+θ)(0＜θ＜π) is an even function, then f(x) is a monotonically increasing interval on [0,π] yes()

A.0,π2 B.π2,π

C.π4,π2 D.3π4,π

(2) The known function f(x)=2sin2x+π6+a+1 (where a is a constant).

① Find the monotonic interval of f(x);

②If x∈0, when π2, the maximum value of f(x) is 4, find the value of a.

[Idea Tips] (1) First find θ based on the function f(x) being an even function, then find the increasing interval based on monotonicity, and finally find the intersection with [0,π].

(2)① Find the increasing interval from 2kπ-π2≤2x+π6≤2kπ+π2, k∈Z,

From 2kπ+π2≤2x+π6≤2kπ+3π2, k∈Z finds the subtraction interval.

② First find the maximum value of f(x), get the equation about a, and then find the value of a.

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Update Time: 2024-09-28

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