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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