Day 2: Continue Sum and Difference of Angles
using Sine and Cosine
Topics in Trigonometry
Example 1: Find the exact value of the
expression:
sin 15°
*Re*Re-write the expression using one of the formulas in order to
calculate the exact value using a sum or difference of two angles
we know how to evaluate using special triangles or the unit circle*
sin 15°= sin(45 − 30) = sin 45° cos 30° − cos 45° sin 30°
*Use special right triangles to evaluate the function*
 2  3   2  1 

−
 
= 





 2  2   2  2 
 6  2
−
 =
= 



 4   4 
6− 2
4
Example 2: Find the exact value of the
expression:
cos 75°
*Re*Re-write the expression using one of the formulas in order to
calculate the exact value using a sum or difference of two angles
we know how to evaluate using special triangles or the unit circle*
cos 75 ° = cos(45 + 30) = cos 45° cos 30° − sin 45° sin 30°
*Use special right triangles to evaluate the function*
 2  3   2  1 

−
 
= 





 2  2   2  2 
 6  2
−
 =
= 



 4   4 
Sum and Difference of Angles
6− 2
4
1
Day 2: Continue Sum and Difference of Angles
using Sine and Cosine
Topics in Trigonometry
Example 3: Find the exact value of the
expression:
sin 105°
*Re*Re-write the expression using one of the formulas in order to
calculate the exact value using a sum or difference of two angles
we know how to evaluate using special triangles or the unit circle*
sin 105 °= sin(60 + 45) = sin 60° cos 45° + cos 60° sin 45°
*Use special right triangles to evaluate the function*
 3  2   1  2 

 +  

= 





 2  2   2  2 
 6  2
+
 =
= 



 4   4 
6+ 2
4
Example 4: Write an expression that is
equivalent to the given expression:
sin(90° − θ )
*Use the correct formula to
expand the expression.
= sin(90° − θ ) = sin 90° cos θ − cos 90° sin θ
*Evaluate the expressions, where possible*
= (1)(cos θ ) − (0)(sin θ )
*Simplify*
= cos θ
Sum and Difference of Angles
2
Day 2: Continue Sum and Difference of Angles
using Sine and Cosine
Topics in Trigonometry
Example 5: Write an expression that is
equivalent to the given expression:
cos(2π − θ )
*Use the correct formula to
expand the expression.
= cos(2π − θ ) = cos(2π ) cos θ + sin(2π ) sin θ
*Evaluate the expressions, where possible*
= cos(360°) cos θ + sin(360°) sin θ
= (1)(cos θ ) + (0) sin θ = cos θ
*Simplify*
Example 6: Write an expression that is
equivalent to the given expression:
sin(x + π )
*Use the correct formula to
expand the expression.
sin(x + π ) = sin x cos π + cos x sin π
*Evaluate the expressions, where possible*
= sin x cos 180° + cos x sin 180°
= sin x ( −1) + cos x (0) *Simplify*
= − sin x
Sum and Difference of Angles
3