Sum and Difference Identities

Sum and Difference IdentitiesUsing the sum and difference identities for sine, cosine, and tangent functions

Sum and Difference Identities for the Cosine FunctionIf α and β represent the

measures of two angles, then the following identities hold for all values of α and β.

Find cos 15⁰ from values of functions of 30⁰ and 45⁰.

¿𝑐𝑜𝑠 45 °𝑐𝑜𝑠30 °+𝑠𝑖𝑛 45 ° 𝑠𝑖𝑛30 °¿ √22∗ √32

+ √22∗ 12

¿ √6+√24

𝑐𝑜𝑠15 ° ≈0.9659

Find from values of functions of .

Terri Cox is an electrical engineer designing a three-phase AC-generator. Three-phase generators produce three currents fo electricity at one time. They can generate more power for the amount of materials used and lead to better transmission and use of power then single-phase generators can. The three phases of the generator Ms. Cox is making are expressed as, . She must show that each phase is equal to the sum of the other two phases but opposite in sign. To do this, she will show that .

0=0

Sum and Difference Identities for Sine FunctionIf α and β represent the

measures of two angles, then the following identities hold for all values of α and β.

Find sin 75⁰ from values of functions of 30⁰ and 45⁰.

¿ 12∗ √22

+ √22∗ √32

¿ √24

+ √64

𝑠𝑖𝑛75 °≈ .9659

Find sin 15⁰ from values of functions of 30⁰ and 45⁰.

Sum and Difference Identities for the Tangent FunctionIf α and β represent the

measures of two angles, then the following identities hold for all values of α and β.

Find tan 15⁰ from values of functions of 45⁰ and 30⁰

¿1− √3

3

1+ √33

¿

3−√33

3+√33

¿ 3−√33+√3

3−√33−√3

=9−6√3+39−3

¿ 12−6√36

=2−√3≈0.2679

Find tan 105⁰ from values of functions of 45⁰ and 60⁰

tan 105=¿tan 45+ tan 601− tan 45 tan 60 ¿

¿ 1+√31−(1)(√3)¿

1+√31−√3

1+√31+√3

=1+2√3+31−3

¿ 4+2√3−2

=−2−√3 ≈−3.7321

Verify that cot 𝑥= tan( 𝜋2 −𝑥)cot 𝑥=

sin (𝜋2 −𝑥 )cos ( 𝜋2 −𝑥)

cot 𝑥=sin 𝜋2 𝑐𝑜𝑠𝑥− cos

𝜋2 sin 𝑥

cos 𝜋2cos 𝑥+sin 𝜋

2sin 𝑥

cot 𝑥=(1 ) cos 𝑥− (0 ) sin𝑥(0 ) cos 𝑥+ (1 )sin 𝑥

cot 𝑥=cos 𝑥sin 𝑥

cot 𝑥=cot𝑥