"Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware

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"Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware

Part One: Explanation of Curriculum Standards

1. Be able to understand the definition of trigonometric functions with the help of the unit circle and the plane rectangular coordinate system. Be able to find the trigonometric function value of a given angle.

2. Be proficient in the sign rules of trigonometric functions in each quadrant, and be able to determine the sign of the trigonometric function value for a given angle.

3. Master the definition domains of sine, cosine and tangent functions.

4. Master the induction formula 1 and be able to use the formula to solve related problems.

PPT on the concept of trigonometric functions, part 2: independent preview

1. Definition of trigonometric functions

1. In the Cartesian coordinate system, the circle with the origin O as the center and unit length as the radius is called the unit circle.

As shown in the figure, if the intersection point of the terminal side of an acute angle α and the unit circle is P (x, y), according to the definitions of sine, cosine, and tangent in right triangles learned in junior high school, can you use the coordinates of point P to express sin α , cos α, tan α? Can this conclusion be extended to the case when α is any angle?

2. Fill in the blanks

As shown in the figure, α is an arbitrary angle, with the vertex O of α as the coordinate origin, and the starting edge of α as the positive half-axis of the x-axis, a plane rectangular coordinate system is established.

Let P(x,y) be the intersection point of the terminal side of α and the unit circle.

(1) The ordinate y of point P is called the sine function of α, recorded as sin α, that is, y=sin α;

(2) The abscissa x of point P is called the cosine function of α, and is recorded as cos α, that is, x=cos α;

(3) The ratio of the ordinate and the abscissa of point P is called the tangent of α, and is recorded as tan α, that is, =tan α(x≠0).

Sine, cosine, and tangent are all functions with angle as the independent variable and the coordinates or ratio of coordinates of points on the unit circle as the function value. We collectively call them trigonometric functions.

3. Fill in the blanks

If there is a point M(x,y) on the terminal side of angle α, and the distance from M to the origin is r=√(x^2+y^2), then sin α=y/r, cos α=x/r, tan α=y/x.

The concept of trigonometric functions PPT, the third part of the content: inquiry learning

Use the definition of trigonometric functions to find the values ​​of trigonometric functions

Example 1 Solve the following problems:

(1) If the intersection point of the terminal side of angle α and the unit circle is P(x"," 2/3), then sin α=_________, cos α=_________, tan α=_________.

(2) It is known that the terminal side of angle α passes through point P(-x,-6), and cos α=-5/13, then 1/sinα+1/tanα=_________.

(3) It is known that the starting side of the angle α coincides with the non-negative semi-axis of the x-axis, and the terminal side is on the ray 4x-3y=0 (x≤0), then cos α-sin α=_________.

Analysis: (1) First find the value of x, and then calculate; (2) Use the generalization of the definition of trigonometric functions to solve; (3) First pick a point on the terminal edge, and then use the definition to solve.

(1) Analysis: According to the meaning of the question, x2+(2/3)^2=1, the solution is x=±√5/3, so sin α=2/3, cos α=±√5/3, tan α= (2/3)/(±√5/3)=±(2√5)/5.

Answer: 2/3　±√5/3　±(2√5)/5

(2) Analysis: The terminal side of angle α passes through point P(-x,-6), and cos α=-5/13, ∴cos α=("-" x)/√(x^2+36) =-5/13, the solution is x=5/2,

∴P -5/2,-6 ,∴sin α=-12/13,

∴tan α=12/5,

Then 1/sinα+1/tanα=-13/12+5/12=-2/3.

Answer:-2/3

The concept of trigonometric functions PPT, the fourth part: thinking analysis

Ignoring the classification discussion of parameters leads to mistakes

Typical example: The terminal side of angle α passes through point P(-3a,4a), a≠0, then cos α=______.

The wrong solution is because x=-3a, y=4a, so r=√("(-" 3a")" ^2+"(" 4a")" ^2 )=5a, so cos α=("-" 3a )/5a=-3/5.

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

Tip: In the wrong solution, I mistakenly thought that a>0, and did not classify and discuss the positive and negative of a, which led to the wrong calculation of r and thus the wrong result.

Correct answer: From the meaning of the question, we can get |OP|=√("(-" 3a")" ^2+"(" 4a")" ^2 )=5|a|, and a≠0.

When a>0, |OP|=5a, then cos α=("-" 3a)/5a=-3/5.

When a<0, |OP|=-5a, then cos α=("-" 3a)/("-" 5a)=3/5.

Answer:-3/5 or 3/5

The concept of trigonometric functions PPT, part 5: practice in class

1. It is known that sin α=5/13, cos α=-12/13, then the coordinates of the intersection of the terminal side of angle α and the unit circle are ()

A.(5/13 ",-" 12/13) B.("-" 5/13 "," 12/13)

C.(12/13 ",-" 5/13) D.("-" 12/13 "," 5/13)

Analysis: From the definition of trigonometric functions, it is easy to obtain that the coordinates of the intersection of the terminal side of angle α and the unit circle are ("-" 12/13 "," 5/13)

Answer:D

2. The terminal side of the known angle θ passes through the point (4,-3), then cos θ=()

A.4/5 B.-4/5 C.3/5 D.-3/5

Analysis: It is known that x=4, y=-3, so r=√(4^2+"(-" 3")" ^2 )=5, so cos θ=4/5.

Answer:A

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Update Time: 2024-07-29

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