1. Chapter 6: Trigonometry 6.5: Basic Trigonometric Identities
Essential Question: What is the Pythagorean Identity? How can it be used to find other trigonometric identities?

2. 6.5: Basic Trigonometric Identities
Convert to radian mode for this section
2nd → more
Parenthesis matter
“cos (x + 3)” and “cos x + 3” yield different results
Why?
(cos t)2 is written (on paper) as cos2 t, because we’re squaring the result of the cosine function, not the “t”

3. 6.5: Basic Trigonometric Identities
Quotient Identities
Example 1: Simplify the expression: tan t cos t
The key for conversion is to get everything in terms of sin and cos.

4. 6.5: Basic Trigonometric Identities
Reciprocal Identities
We’ve seen these before
Example 2: Reciprocal Identities
Given that sin t = 0.28 and cos t = Find csc t and sec t

5. 6.5: Basic Trigonometric Identities
Pythagorean Identities
You know the traditional, a2 + b2 = c2
sin2 t + cos2 t = 1
The other two identities can be derived from this equation
tan2 t + 1 = sec2 t
1 + cot2 t = csc2 t
Example 3: Using Pythagorean identities
Simplify the equation: tan2 t cos2 t + cos2 t
Rewrite using just sin & cos

6. 6.5: Basic Trigonometric Identities
Example 4: Finding all other values from one
The value for trigonometric function is given for 0 < t < π/2. Use quotient, reciprocal and Pythagorean identities to find the other values of the remaining five trigonometric functions. Round your answers to four decimal places.
sec t = cos t = ?
csc t = ? sin t = ?
cot t = ? tan t = ?

7. 6.5: Basic Trigonometric Identities
sec t = cos t =
csc t = ? sin t = ?
cot t = ? tan t =
cos can be found as it’s the reciprocal of sec
cos t = 1/sec t = 1/ =
tan can be found with the Pythagorean theorem
sec2 t = tan2 t + 1
– 1 = tan2 t
= tan t

8. 6.5: Basic Trigonometric Identities
sec t = cos t =
csc t = sin t =
cot t = tan t =
cot can be found as it’s the reciprocal of tan
cot t = 1/tan t = 1/ =
tan t = sin t / cos t
= sin t /
● = sin t
= sin t
csc can be found by taking the reciprocal of sin
csc t = 1/sin t = 1/ =

9. 6.5: Basic Trigonometric Identities
Assignment
Page 460
1 – 25, odd problems
Show work