MHF4U NAME: ________________________ DATE: ______________
7.2 – 7.4 CLASS NOTES COMPOUND & DOUBLE ANGLE FORMULAS PROVING TRIGONOMETRIC IDENTITIES
7.2 – COMPOUND ANGLE FORMULAS A compound angle is made by __________________or ___________________ two or more angles. There are special formulas called compound angle formulas that help us determine the EXACT values of sums or differences of angles. Let’s prove some of these formulas! PRACTICE EXAMPLE 1: Prove 𝐜𝐨𝐬 𝒂− 𝒃 = 𝐜𝐨𝐬𝒂 𝐜𝐨𝐬𝒃+ 𝐬𝐢𝐧𝒂 𝐬𝐢𝐧𝒃 1) Recall: Cosine Law:
2) Recall: Distance Formula:
3) Set equations equal to each other: PRACTICE EXAMPLE 2: Prove 𝒄𝒐𝒔 𝒂+ 𝒃 = 𝒄𝒐𝒔 𝒂 𝒄𝒐𝒔 𝒃− 𝒔𝒊𝒏 𝒂 𝒔𝒊𝒏 𝒃 PRACTICE EXAMPLE 3: Given 𝒔𝒊𝒏 𝒂+ 𝒃 = 𝒄𝒐𝒔 𝒂 𝒔𝒊𝒏 𝒃+ 𝒔𝒊𝒏 𝒂 𝒄𝒐𝒔 𝒃, use the quotient identity, 𝒕𝒂𝒏𝜽 = 𝒔𝒊𝒏𝜽
𝒄𝒐𝒔𝜽 , to determine a compound angle formula for 𝒕𝒂𝒏 𝒂+ 𝒃 .
SUMMARY: COMPOUND (ADDITION/SUBTRACTION) FORMULAS:
𝐬𝐢𝐧 𝒙 + 𝒚 = 𝒔𝒊𝒏 𝒙 − 𝒚 = 𝒄𝒐𝒔 𝒙 + 𝒚 = 𝒄𝒐𝒔 𝒙 − 𝒚 = 𝒕𝒂𝒏 𝒙 + 𝒚 = 𝒕𝒂𝒏 𝒙 − 𝒚 =
PRACTICE EXAMPLE 4: Write the following as single trig ratios. a) b) PRACTICE EXAMPLE 5: Write the following as single trig ratios and then evaluate. a) b)
PRACTICE EXAMPLE 6: Determine the EXACT value of: a) b)
7.2 HOMEWORK: Pg. 400 # 1, 2, 3ace, 4ace, 5ace, 6ace, 9abcd, 11, 12a
7.3 – DOUBLE ANGLE FORMULAS Explore: Using the compound angle formulas, develop formulas for sin 2𝜃 (𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 sin𝑎𝑛𝑑 cos ), cos 2𝜃 (in terms of sin and cos, in terms of cos, and in terms of sin), tan 2𝜃 (in terms of tan). using the compound angle formulas from 7.2. (Recall the Pythagorean identity: sin! 𝜃 + cos! 𝜃 = 1. Also, use 𝜃 for both A & B). SUMMARY: DOUBLE ANGLE FORMULAS:
𝐬𝐢𝐧𝟐𝜽
𝐜𝐨𝐬𝟐𝜽 𝐭𝐚𝐧𝟐𝜽
PRACTICE EXAMPLE 1: Express as a single trig function. a) b)
c) d) PRACTICE EXAMPLE 2: If 𝒕𝒂𝒏 𝒙 = 𝟐
𝟓,𝝅 ≤ 𝒙 ≤ 𝟐𝝅, find the exact value of:
a) cos 2𝑥 b) sin 2𝑥 c) tan 2𝑥
Explore: Develop a formulae for cos !
!. (Hint: use 𝑥 = !
! 𝑖𝑛 cos 2𝑥 = 2 cos! 𝑥 − 1 )
7.3 HOMEWORK: Pg. 407 # 1, 2a-‐d, 3abc, 4, 6, 7, 8, 11a 7.4 – PROVING TRIG IDENTITIES What is an “identity?” It is a mathematical statement that is true for all values of the given variable. For example, 3𝑥 + 2 + 5𝑥 = 2 4𝑥 + 1 “Proving” Identities:
• Identities are considered to be “proven” if we can show that the _________________ and __________________ of the identity simplify to the same expression.
• How you PRESENT your proof is VERY IMPORTANT! Anyone reading your proof must be able to follow what you are doing and agree that your work is valid. Overall,
__________________________________________!!!
How to Prove a Trigonometric Identity: 1) Choose the more “complex looking” side to simplify first. Keep working on this side until it is fully simplified or until you get stuck à ONLY THEN start working on the other side. 2) Try to write all expressions in terms of SIN and COS, unless better method obvious. 3) Use known simple trig identities (below) to simplify the expressions. 4) Apply algebraic manipulation when possible.
• Multiply top and bottom by a common denominator. • Factor numerators and denominators, and cancel out factors (Reduce!). • You may have to deal with “fractions divided by fractions”. • Multiplying by the conjugate expression may help.
5) When the left side equals the right, write 𝐿𝑆 = 𝑅𝑆, ∴ 𝑄𝐸𝐷 as your statement. Some very useful Trig Identities: You will be referring to these “simple” identities while proving more complex ones. You already know most of these! The more identities you attempt to solve, the better you will be at it! Therefore, ________________________!!!
PRACTICE EXAMPLE 1: Prove: a) b)
d) !"#! !!!"#! !!"#! !!!"#! !
= 1 c) 7.4 HOMEWORK: 1) Proving Trig Identities WS (Attached) 2) Pg. 417 #9abc, 10abcd, 11abdik MHF4U NAME: _____________________ DATE: _______________
PROVING TRIGONOMETRIC IDENTITIES
Prove the following trig identities. Have the simple trig formulas available on your desk to refer to. Solutions can be found on the course website.
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