6.4 TRIGONOMETRIC ANGLES



HOMEWORK: Sec 2.2: 3-10, 13-16, 18, 19, 23, 24, 28-33,

37-54, 57-60

Read Sec 2.3

sin( = y / r = opp / hyp

cos ( = x / r = adj / hyp

tan ( = y / x = opp / adj (x ( 0)

csc ( = r / y = hyp / opp (y ( 0)

sec ( = r / x = hyp / adj (x ( 0)

cot ( = x / y = adj / opp (y ( 0)

SOHCAHTOA

I. Finding the values of the six trig function of a triangle

(

3

(

2

II. Given the value of one trig function, find the others. Assume all angles are QI.

Example 1: [pic]

Example 2: [pic]

FUNDAMENTAL IDENTITIES:

**** Very Important to MEMORIZE these ASAP ****

RECIPROCAL IDENTITIES:

[pic]

TANGENT/COTANGENT QUOTIENT:

[pic]

PYTHAGOREAN IDENTITIES:

[pic]

Example: Use trig identities to find the values of the other trig functions if sin ( = ½ if θ in QI

Example (to try at home): Use trig identities to find the values of the other trig functions if tan ( = ½ if θ in QI

Work the problem again using the triangle method:

Answer: [pic]

Avoid common mistakes: Note: tan ( = ½ DOES NOT MEAN

sin θ = 1 and cos θ = 2! The denominators canceled! You need to use Pythagorean ID’s!

Example: Use trig identities to verify the following identities

[pic]

[pic]

Watch for common mistakes: Never drop variables! I.e. cos is not the same as cos θ!

COMPLEMENTARY: Two angles are complementary if their sum is [pic]or 90º

Example: Find the complement of [pic]

SUPPLEMENTARY: Two angles are supplementary if their sum is [pic]or 180º

Example: Find the supplement of [pic]

** Important Note: You MUST leave answer in the same angle mode as the original angle unless directed to convert.

Complementary Angle Theorem

Co-functions of complementary angles are equal

sin θ = cos(90º- θ) or sin θ = [pic]

cos θ = sin(90º- θ) or cos θ = [pic]

tan θ = cot(90º- θ) or tan θ = cot[pic]

cot θ = tan(90º- θ) or cot θ = tan[pic]

csc θ = sec(90º- θ) or csc θ = sec[pic]

sec θ = csc(90º- θ) or sec θ = csc[pic]

Examples: cos(40º)=sin(50º) because

tan(30º)=cot(60º) because

sec(25º)=csc(65º) because

[pic] because

Example: Use identities to simplify the following without a calculator

[pic]

[pic]

Note: You MUST have all ID’s MEMORIZED and practice enough problems to be able to recognize them quickly!

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