HPC Ch.5 Intro Activity

HPC Ch.5 Intro Activity

Trigonometric Identities

Objective: To study Reciprocal, Even & Odd, Pythagorean, Quotient identities (you will be tested in the following slides TODAY since we will use them to do proofs, but we can not use them unless we prove them first). Finish all the slides during class, then you don't have HW today.

What is Trig Identities?

• Trig Identities are equalities that involve trig functions and are true for all values of the defined variable.

• Reciprocal Identities

• Even & Odd Identities

• Pythagorean Identities

• Quotient Identities and much MUCH more!

• Since they are true for all values, we prove them 1st, and then use them in our proofs.

Reciprocal Identities

•  $\sin\ x\ =\ \frac{1}{\csc\ x}$  

•  $\cos\ x\ =\ \frac{1}{\sec\ x}$  

•  $\tan\ x\ =\ \frac{1}{\cot\ x}$  

•  $\csc\ x\ =\ \frac{1}{\sin\ x}$  

•  $\sec\ x\ =\ \frac{1}{\cos\ x}$  

•  $\cot\ x\ =\ \frac{1}{\tan\ x}$  

Multiple Select

Which statements are true?

•  $\sin\ x\ =\ \frac{1}{\cos\ x}$  

•  $\cos x\ =\ \frac{1}{\sec\ x}$  

•  $\tan\ x\ =\ \frac{1}{\cos\ x}$  

•  $\csc\ x\ =\ \frac{1}{\sin\ x}$  

•  $\sec\ x\ =\ \frac{1}{\cos\ x}$  

Even-Odd Identities (Remember that only cos & sec is even identities) Even function means f(x) = f(-x) or symmetry respect to y-axis.

• sin(-x) = - sin x

• cos(-x) = cos x

• tan(-x) = - tan x

• csc(-x) = - csc x

• sec(-x) = sec x

• cot (-x) = - cot x

Even functions

f(x) = f(-x) or symmetry respect to y - axis

Odd function means f(-x) = - f(x) or symmetry respect to the origin.

• sin(-x) = - sin x

• tan(-x) = - tan x

• csc(-x) = - csc x

• cot (-x) = - cot x

Multiple Select

Which statements are true?

• sin ( - x) = - sin x

• cos (-x) = cos x

• tan (-x) = tan x

• csc ( - x) = csc x

• cot ( - x) = - cot x

Pythagorean Identities

• sin² x + cos² x = 1. Notice sin² x is equivalent as (sin x)(sin x)

• 1 + tan² x = sec² x

• 1 + cot² x = csc² x

• Study 3 Pythagorean Identities for 5 min.

Multiple Choice

sin² x + cos² x = ?

• 1

• 0

• -1

• 2

• cot²x

Multiple Choice

1 + tan² x = ?

• sec² x

• cos² x

• sin² x

• cot² x

Multiple Choice

1 + cot² x = ?

• sin² x

• cos² x

• csc² x

• tan² x

Multiple Choice

1 - cos² x = ??? if sin² x + cos² x = 1

• sin² x

• cos² x

• tan² x

• cot² x

Multiple Choice

1 - sin² x = ??? if sin² x + cos² x = 1

• sin² x

• cos² x

• tan² x

• sec² x

Multiple Choice

sin² x - 1 = ??? if sin² x + cos² x = 1

• sin² x

• cos² x

• tan² x

• sec² x

• -cos² x

Multiple Choice

sec² x - 1 = ??? if 1 + tan² x = sec² x

• sin² x

• cos² x

• tan² x

• sec² x

• -cos² x

Multiple Choice

csc² x - 1 = ??? if 1 + cot² x = csc² x

• sin² x

• cos² x

• tan² x

• sec² x

• cot² x

Multiple Choice

sec² x - tan² x = ??? if 1 + tan² x = sec² x

• sin² x

• cos² x

• 1

• -1

• cot² x

Multiple Choice

csc² x - cot² x = ??? if 1 + cot² x = csc² x

• sin² x

• cos² x

• -1

• 1

• cot² x

Multiple Choice

cot² x - csc²x = ??? if 1 + cot² x = csc² x

• sin² x

• cos² x

• -1

• 1

• cot² x

Quotient Identities

•  $\tan\ x\ =\ \frac{\sin\ x\ }{\cos\ x}$  

•  $\cot\ x\ =\ \frac{\cos\ x}{\sin\ x}$  

• Sometimes we use  $\tan^2x\ =\ \frac{\sin^2x}{\cos^2x}$  

• Or  $\cot^2x=\frac{\cos^2x}{\sin^2x}$