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Sum and Difference Formulas. Dr .Hayk Melikyan Departmen of Mathematics and CS [email protected] The Cosine of the Difference of Two Angles. The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the

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Sum and difference formulas

Sum and Difference Formulas

Dr .Hayk Melikyan

Departmen of Mathematics and CS

[email protected]


The cosine of the difference of two angles
The Cosine of the Difference of Two Angles

The cosine of the difference of two angles equals the

cosine of the first angle times the cosine of the

second angle plus the sine of the first angle times the

sine of the second angle.


Text example

Solution We know exact values for trigonometric functions of 60° and 45°. Thus, we write 15° as 60° - 45° and use the difference formula for cosines.

cos l5° = cos(60° - 45°)

= cos 60° cos 45° + sin 60° sin 45°

cos(-) = cos  cos + sin  sin 

Text Example

  • Find the exact value of cos 15°

Substitute exact values from

memory or use special triangles.

Multiply.

Add.


Text example1

cos(-) = cos  cos + sin  sin 

Text Example

Find the exact value of ( cos 80° cos 20° + sin 80° sin 20°) .

Solution The given expression is the right side of the formula for cos( - ) with = 80° and  = 20°.

cos 80° cos 20° + sin 80° sin 20° = cos (80° - 20°) = cos 60° = 1/2


Example
Example

  • Find the exact value of cos(180º-30º)

Solution


Example1
Example

  • Verify the following identity:

Solution


Sum and difference formulas for cosines and sines
Sum and Difference Formulas for Cosines and Sines


Example2
Example

Solution

  • Find the exact value of sin(30º+45º)


Sum and difference formulas for tangents
Sum and Difference Formulas for Tangents

The tangent of the sum of two angles equals the tangent of the first angle plus the tangent of the second angle divided by 1 minus their product.

The tangent of the difference of two angles equals the tangent of the first angle

minus the tangent of the second angle divided by 1 plus their product.


Example3
Example

Solution

  • Find the exact value of tan(105º)

  • tan(105º)=tan(60º+45º)


Example4
Example

  • Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

Solution


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