Sum and Difference Formulas Section 5.4

Exploration:  Are the following functions equal? a) Y = Cos (x + 2)b) Y = Cos x + Cos 2 How can we determine if they are equal by looking at their graphs? Graph them using your calculator.

Exploration a) Y = Cos (x + 2)b) Y = Cos x + Cos 2 Y = Cos (x + 2) Y = Cos x + Cos 2 Y = Cos (x + 2)

Sum and Difference Formulas  Sin (u + v) =  Sin (u – v) =  Cos (u + v) =  Cos (u – v) = Sin u Cos v + Cos u Sin v Sin u Cos v – Cos u Sin v Cos u Cos v – Sin u Sin v Cos u Cos v + Sin u Sin v

Sum and Difference Formulas  Tan (u + v) =  Tan (u – v) = Tan u + Tan v 1 - Tan u Tan v Tan u - Tan v 1 + Tan u Tan v

Sum and Difference Formulas  Before we continue, think about all of the angles you can find a trig function without using a calculator: Choose any 2 of these and a trig function:

Sum and Difference Formulas

 To find the trig function of an angle using the formulas: 1) Find 2 angles whose sum or difference is equal to the angle you are trying to evaluate 2) Put the two angles into the appropriate formula 3) Evaluate the trig functions of the angles you know 4) Simplify

Sum and Difference Formulas  Evaluate: Sin 15 º What two angles have a sum or difference of 15º? → 45º - 30º Put these two angles in the appropriate formula: → Sin (45º - 30º) = Sin 45 º Cos 30º - Cos 45º Sin 30º

Sum and Difference Formulas Sin 45 º Cos 30º - Cos 45º Sin 30º Evaluate the trig functions Simplify

Sum and Difference Formulas

 Evaluate the following functions. a) b)

Sum and Difference Formulas = Cos 45 º Cos 30º - Sin 45º Sin 30º

Sum and Difference Formulas

Section 5.4

Sum and Difference Formulas  Evaluate the following functions. a) b)

Sum and Difference Formulas = Cos 150 º Cos 45º + Sin 150º Sin 45º

Sum and Difference Formulas

 Yesterday:  Used the formulas to evaluate trig functions of different angles  Worked with both radians and degrees  Today  Use the formulas to simplify longer expressions  Use the formulas to evaluate expressions from triangles  Use the formulas to create algebraic expressions

Sum and Difference Formulas  Find the exact value of the following expression: Cos 78 ºCos18º + Sin 78ºSin18º What formula is being used here? → Cos (u – v) Re-write the expression using the formula → Cos (78º – 18º) = Cos 60º= ½

Sum and Difference Formulas  Use the sum and difference formulas to evaluate the following:

Sum and Difference Formulas  Find the exact value of the Cos (u – v) using the given information: Sin u = Cos v = Both u and v and in quadrant III When you are given 2 different criteria, you must draw 2 different triangles u v

Sum and Difference Formulas u v Cos (u – v) =Cos u Cos v + Sin u Sin v

Sum and Difference Formulas  Find the exact value of the trig functions given the following information: Tan u = Csc v = and both u and v are in quadrant IV. Find a) Sin (u + v) b) Sec (u – v) c) Cot (u – v)

Sum and Difference Formulas u v Sin (u + v) =Sin u Cos v + Cos u Sin v

Sum and Difference Formulas u v Sec (u - v) = Cos u Cos v + Sin u Sin v 1 ÷ Cos (u - v) Cos (u - v) =

Sum and Difference Formulas u v Cot (u - v) = 1 ÷ Tan (u - v) Tan (u - v) =

Sum and Difference Formulas  Lastly, we would like to apply the process used in drawing triangles to create algebraic expressions.  Same steps as before, just using variables instead of numbers.

Sum and Difference Formulas  Write Cos (arcTan 1 + arcCos x) as an algebraic statement. → What formula is being used? Cos (u + v) u = arcTan 1v = arcCos x Tan u = 1Cos v = x → Use this information to draw your triangles.

Sum and Difference Formulas Tan u = 1Cos v = x u 1 1 v x 1 Cos (u + v) = Cos u Cos v – Sin u Sin v

Sum and Difference Formulas  Write the trig expression as an algebraic expression: Sin (arcTan 2x – arcCos x) Sin (u – v) u = arcTan 2x v = arcCos x Tan u = 2x Cos v = x

Sum and Difference Formulas u 2x 1 v x 1 Sin (u – v) = Sin u Cos v – Cos u Sin v Tan u = 2x Cos v = x

Sum and Difference Formulas Section 5.4

Sum and Difference Formulas  Write the trig expression as an algebraic expression: Cos (arcSin 3x + arcTan 2x) Cos (u + v) u = arcSin 3x v = arcTan 2x Sin u = 3x Tan v = 2x

Sum and Difference Formulas u 3x 1 v 1 Cos (u + v) = Cos u Cos v – Sin u Sin v Sin u = 3x Tan v = 2x

Sum and Difference Formulas  So far, in this section we have: a) Used sum and difference formulas to evaluate trig functions of different angles b) Recognized sum and difference formulas to simplify expressions c) Used criteria to draw triangles and apply formulas d) Create algebraic expressions Lastly, we are going to simplify, verify, and solve equations

Sum and Difference Formulas  Simplifying: o Apply the formula o Evaluate trig functions that you know o Reduce the expression

Sum and Difference Formulas  Simplify the following expressioni: Sin (90 º – x) → Sin 90º Cos x – Cos 90º Sin x → (1)(Cos x)- (0)(Sin x) = Cos x

Sum and Difference Formulas  Simplify the following expressioni: Cos (x + 3 π ) → Cos x Cos 3 π – Sin x Sin 3 π → (Cos x)(0)- (Sin x)(1) = Sin x

Sum and Difference Formulas  Verifying  Same process and simplifying  You are given what the expression should simplify to  As before, only work with 1 side of the equal sign

Sum and Difference Formulas  Verify the following identities: a) Tan ( π + x) = Tan x b) Sin (x + y) Sin (x – y) = Cos² y – Cos² x

Sum and Difference Formulas Tan ( π + x) = Tan x

Sum and Difference Formulas Sin (x + y) Sin (x – y) = Cos² y – Cos² x = (Sin x Cos y + Cos x Sin y) ( Sin x Cos y – Sin x Cos y) = Sin² x Cos² y - Cos² x Sin² y = (1 - Cos² x) Cos² y - Cos² x (1 – Cos² y) = Cos² y - Cos² x Cos² y- Cos² x + Cos² y Cos² x = Cos² y – Cos² x

Sum and Difference Formulas  The last step in this section is using the sum and difference formulas to solve equations.  Again, apply the formula, simplify, and now solve.

Sum and Difference Formulas