H.Melikyan/12001 Sum and Difference Formulas Dr.Hayk Melikyan Departmen of Mathematics and CS

H.Melikyan/12002 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.

H.Melikyan/12003 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(   )  cos  cos   sin  sin   cos 60° cos 45°  sin 60° sin 45° Substitute exact values from memory or use special triangles. Multiply. Add. Text Example v Find the exact value of cos 15°

H.Melikyan/12004 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 cos(   )  cos  cos   sin  sin  Text Example

H.Melikyan/12005 Example v Find the exact value of cos(180º-30º) Solution

H.Melikyan/12006 Example v Verify the following identity : Solution

H.Melikyan/12007 Sum and Difference Formulas for Cosines and Sines

H.Melikyan/12008 Example v Find the exact value of sin(30º+45º) Solution

H.Melikyan/12009 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.

H.Melikyan/ Example v Find the exact value of tan(105º) tan(105º)=tan(60º+45º) Solution

H.Melikyan/ Example v Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. Solution