Sum and Difference Formulas

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 + Cos 2 Y = Cos (x + 2) Y = Cos (x + 2) Y = Cos x + Cos 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

Sum and Difference Formulas • To find the trig function of an angle using the formulas: • Find 2 angles whose sum or difference is equal to the angle you are trying to evaluate • Put the two angles into the appropriate formula • Evaluate the trig functions of the angles you know • Simplify

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

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

Sum and Difference Formulas • Evaluate the following functions.

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

Sum and Difference Formulas • Evaluate the following functions.

Sum and Difference Formulas = Cos 150º 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 78ºCos18º + Sin 78ºSin18º What formula is being used here? → Cos (u – v) Re-write the expression using the formula → Cos (78º – 18º) = ½ = 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 -24 - 4 u v -7 - 3 25 5

Sum and Difference Formulas -24 - 4 u v -7 - 3 25 5 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 4 12 u v -3 -5 5 13 Sin (u + v) = Sin u Cos v + Cos u Sin v

Sum and Difference Formulas 4 12 u v -3 -5 5 13 1 ÷ Cos (u - v) Sec (u - v) = Cos u Cos v + Sin u Sin v Cos (u - v) =

Sum and Difference Formulas 4 12 u v -3 -5 5 13 1 ÷ Tan (u - v) Cot (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 1 v = arcCos x Tan u = 1 Cos v = x → Use this information to draw your triangles.

Sum and Difference Formulas Tan u = 1 Cos v = x 1 1 u 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) v = arcCos x u = arcTan 2x Cos v = x Tan u = 2x

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

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

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

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

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

Sum and Difference Formulas • Simplify the following expressioni: Sin (90º – 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 + 3π) → Cos x Cos 3π – Sin x Sin 3π → (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: • Tan (π + x) = Tan x • 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 – Sin x Cos y) = (Sin x Cos y + Cos x Sin y) = Sin² x Cos² y - Cos² x Sin² y - Cos² x (1 – Cos² y) = (1 - Cos² x) 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