"The Cosine Formula of the Difference of Two Angles" Trigonometric Functions PPT

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"The Cosine Formula of the Difference of Two Angles" Trigonometric Functions PPT

Part One: Explanation of Curriculum Standards

1. Be able to use the definition of trigonometric functions and the distance formula to derive the cosine formula of the difference between two angles.

2. Master the cosine formula of the difference between two angles and be able to use the formula to solve related problems.

3. Understand the transformation relationship between general and special in the application of formulas.

Cosine formula of the difference between two angles PPT, part 2 content: independent preview

Cosine formula for the difference between two angles

1. Is the 15° angle a special angle? If it is not a special angle, can the sum and difference of special angles be used to express 15°? If 15°=45°-30°, then cos 15°=cos(45°-30° )=cos 45°-cos 30°?

Tip: The 15° angle is not a special angle, but the difference of special angles can be used to represent 15°. For example, 15°=45°-30°, but cos(45°-30°)≠cos 45°-cos 30°.

2. Observe the data in the table below, what do you find?

Tip: cos(60°-30°)=cos 60°cos 30°+sin 60°sin 30°; cos(120°-60°)=cos 120°cos 60°+sin 120°sin 60°.

3. Fill in the blanks

(1)cos(α-β)=cos αcos β+sin αsin β.

(2) This formula is abbreviated as C(α-β).

(3) Conditions of use: α and β are arbitrary angles.

4. Do it

(1)cos 15°=_______.

(2)cos 75°cos 15°+sin 75°sin 15°=_______.

Analysis: (1)cos 15°=cos(45°-30°)=cos 45°cos 30°+sin 45°sin 30°=√2/2×√3/2+√2/2×1/2 =(√6+√2)/4.

(2)cos 75°cos 15°+sin 75°sin 15°=cos(75°-15°)=cos 60°=1/2.

Answer: (1)(√6+√2)/4　(2)1/2

The cosine formula of the difference between two angles PPT, the third part: exploration and learning

Use the cosine formula of the difference between two angles to solve the angle evaluation problem

Example 1 Find the values ​​of the following expressions:

(1)cos(-375°);

(2)cos 75°cos 15°-sin 75°sin 195°;

(3)cos(α+45°)cos α+sin(α+45°)sin α;

(4)1/2cos 15°+√3/2sin 15°.

Analysis: For (1), you should use the induction formula to convert -375° into an acute angle and then into the difference between two special angles and then use the formula to calculate; for (2), convert sin 195° into -sin 15°, and then apply the formula Calculation; for (3), α+45° can be treated as a whole; for (4), 1/2 and √3/2 should be converted into cos 60° and sin 60° respectively, and then calculated using the formula .

Solution: (1)cos(-375°)=cos 375°=cos 15°

=cos(45°-30°)=cos 45°cos 30°+sin 45°sin 30°

=√2/2×√3/2+√2/2×1/2=(√6+√2)/4.

(2)cos 75°cos 15°-sin 75°sin 195°

=cos 75°cos 15°+sin 75°sin 15°

=cos(75°-15°)=cos 60°=1/2.

(3)cos(α+45°)cos α+sin(α+45°)sin α

=cos [(α+45°)-α]=cos 45°=√2/2.

(4)1/2cos 15°+√3/2sin 15°

=cos 60°cos 15°+sin 60°sin 15°

=cos(60°-15°)=cos 45°=√2/2.

Reflection on the methods and skills of using formula C (α-β) to evaluate

When using the cosine formula of the difference between two angles to solve the evaluation problem of a trigonometric function containing non-special angles, you must first convert the non-special angle into the difference of a special angle (or the difference between the same non-special angle and a special angle), and use The formula is directly simplified and evaluated. During the conversion process, the induced formula is fully utilized to construct the structural form of the cosine formula of the difference between two angles, and the formula can be used correctly or inversely to evaluate.

Cosine formula of the difference between two angles PPT, part 4: practice in class

1.cos 50°=()

A.cos 70°cos 20°-sin 70°sin 20°

B.cos 70°sin 20°-sin 70°cos 20°

C.cos 70°cos 20°+sin 70°sin 20°

D.cos 70°sin 20°+sin 70°cos 20°

Analysis: cos 50°=cos(70°-20°)=cos 70°cos 20°+sin 70°sin 20°.

Answer:C

2.The value of cos5π/12cosπ/6+cosπ/12sinπ/6 is ()

A.0 B.1/2 C.√2/2 D.√3/2

Analysis: cos5π/12cosπ/6+cosπ/12sinπ/6=cos5π/12cosπ/6+sin5π/12sinπ/6=cos(5π/12 "-" π/6)=cosπ/4=√2/2.

Answer:C

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Update Time: 2024-07-29

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