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Fundamental Trigonometric Identities

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Basic Trigonometric Identities

The basic trigonometric identities are ones that can be logically deduced from the definitions and graphs of the six trigonometric functions.  Previously, some of these identities have been used in a casual way, but now they will be formalized and added to the toolbox of trigonometric identities. 

How can you use the trigonometric identities to simplify the following expression?

\left[\frac{\sin \left(\frac{\pi}{2}-\theta\right)}{\sin (-\theta)}\right]^{-1}

Watch This

http://www.youtube.com/watch?v=_gkuml--4_Q  James Sousa: Cofunction Identities

http://www.youtube.com/watch?v=YbU8Sq0quWE James Sousa: Even and Odd Trigonometric Identities

Guidance

The reciprocal identities refer to the connections between the trigonometric functions like sine and cosecant.  Sine is opposite over hypotenuse and cosecant is hypotenuse over opposite.  This logic produces the following six identities.

  • \sin \theta=\frac{1}{\csc \theta}
  • \cos \theta=\frac{1}{\sec \theta}
  • \tan \theta=\frac{1}{\cot \theta}
  • \cot \theta=\frac{1}{\tan \theta}
  • \sec \theta=\frac{1}{\cos \theta}
  • \csc \theta=\frac{1}{\sin \theta}  

The quotient identities follow from the definition of sine, cosine and tangent.

  • \tan \theta=\frac{\sin \theta}{\cos \theta}
  • \cot \theta=\frac{\cos \theta}{\sin \theta}

The odd-even identities follow from the fact that only cosine and its reciprocal secant are even and the rest of the trigonometric functions are odd.

  • \sin (-\theta)=-\sin \theta
  • \cos (-\theta)=\cos \theta
  • \tan (-\theta)=-\tan \theta
  • \cot (-\theta)=-\cot \theta
  • \sec (-\theta)=\sec \theta
  • \csc (-\theta)=-\csc \theta

The cofunction identities make the connection between trigonometric functions and their “co” counterparts like sine and cosine.  Graphically, all of the cofunctions are reflections and horizontal shifts of each other.

  • \cos \left(\frac{\pi}{2}-\theta \right)=\sin \theta
  • \sin \left(\frac{\pi}{2}-\theta \right)=\cos \theta
  • \tan \left(\frac{\pi}{2}-\theta \right)=\cot \theta
  • \cot \left(\frac{\pi}{2}-\theta \right)=\tan \theta
  • \sec \left(\frac{\pi}{2}-\theta \right)=\csc \theta
  • \csc \left(\frac{\pi}{2}-\theta \right)=\sec \theta

Example A

If \sin \theta=0.87 , find \cos \left(\theta- \frac{\pi}{2}\right) .

Solution:  While it is possible to use a calculator to find \theta , using identities works very well too. 

First you should factor out the negative from the argument.  Next you should note that cosine is even and apply the odd-even identity to discard the negative in the argument.  Lastly recognize the cofunction identity.

\cos \left(\theta-\frac{\pi}{2}\right)=\cos \left(-\left(\frac{\pi}{2}-\theta \right)\right)=\cos \left(\frac{\pi}{2}-\theta \right)=\sin \theta=0.87

Example B

Use identities to simplify the following expression: \tan x \cot x+\frac{\sin x \cot x (\sec x)^2}{\sec x} .

Solution:  Start by rewriting the expression and replacing one or two terms that you see will cancel.  In this case, replace the \cot x=\frac{1}{\tan x}  and cancel the secant term.

=\tan x \cdot \frac{1}{\tan x}+\sin x \cot x \sec x

Cancel the tangents to make a one and then use the quotient and reciprocal identities to rewrite the right part of the expression in terms of just sines and cosines.  Lastly you should cancel and simplify.

&= 1 + \sin x \cdot \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}\\&= 1+1\\&= 2

Example C

Use identities to prove the following: \cot (- \beta) \cot \left(\frac{\pi}{2}-\beta \right) \sin (- \beta)=\cos \left(\beta-\frac{\pi}{2}\right) .

Solution:  When doing trigonometric proofs, it is vital that you start on one side and only work with that side until you derive what is on the other side.  Sometimes it may be helpful to work from both sides and find where the two sides meet, but this work is not considered a proof.  You will have to rewrite your steps so they follow from only one side.  In this case, work with the left side and keep rewriting it until you have  \cos \left(\beta - \frac{\pi}{2}\right) .

\cot (-\beta) \cot \left(\frac{\pi}{2}-\beta \right) \sin (- \beta) &= -\cot \beta \tan \beta \cdot - \sin \beta\\&= -1 \cdot -\sin \beta\\&= \sin \beta\\&= \cos \left(\frac{\pi}{2}-\beta \right)\\&= \cos \left(-\left(\beta - \frac{\pi}{2}\right)\right)\\&= \cos \left(\beta-\frac{\pi}{2}\right)

Concept Problem Revisited

The following trigonometric expression can be simplified to be equivalent to negative tangent.

\left[\frac{\sin \left(\frac{\pi}{2}-\theta \right)}{\sin (- \theta)}\right]^{-1} &= \frac{\sin (-\theta)}{\sin \left(\frac{\pi}{2}-\theta \right)}\\&= \frac{-\sin \theta}{\cos \theta}\\&=-\tan \theta

Vocabulary

An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.  In the concept problem, the equivalent expressions are meaningless if the denominator of the rational expression ends up as zero.  This is why identities only work within a valid domain. 

Cofunctions are functions that are identical except for a reflection and horizontal shift.  Examples are sine and cosine, tangent and cotangent, secant and cosecant. 

A proof is a derivation where two sides of an expression are shown to be equivalent through a sequence of logical steps. 

Guided Practice

1. Prove the quotient identity for tangent using the definition of sine, cosine and tangent. 

2. If \cos \left(\theta-\frac{\pi}{2}\right)=0.68  then determine \csc (-\theta) .

3. Prove the following trigonometric identity by working with only one side.

\cos x \sin x \tan x \cot x \sec x \csc x=1

Answers:

1. When tangent, sine and cosine are replaced with the shorthand for side ratios the equivalence becomes a matter of algebra.

\tan \theta &= \frac{opp}{adj}\\&= \frac{\left(\frac{opp}{hyp}\right)}{\left(\frac{adj}{hyp}\right)}\\&= \frac{\sin \theta}{\cos \theta}

2. As Example C and Example A show, \cos \left(\theta-\frac{\pi}{2}\right)=\cos \left(\frac{\pi}{2}-\theta \right) .

0.68 &= \cos \left(\theta-\frac{\pi}{2}\right)\\&= \cos \left(\frac{\pi}{2}-\theta \right)\\&= \sin (\theta)

Then, \csc (-\theta)=-\csc \theta

&= -\frac{1}{\sin \theta}\\&= -(0.68)^{-1}

3. \cos x \sin x \tan x \cot x \sec x \csc x &= \cos x \sin x \tan x \cdot \frac{1}{\tan x} \cdot \frac{1}{\cos x} \cdot \frac{1}{\sin x}\\&= 1

Practice

1. Prove the quotient identity for cotangent using sine and cosine.

2. Explain why \cos \left(\frac{\pi}{2}-\theta \right)=\sin \theta  using graphs and transformations.

3. Explain why \sec \theta=\frac{1}{\cos \theta} .

4. Prove that \tan \theta \cdot \cot \theta=1 .

5. Prove that \sin \theta \cdot \csc \theta=1 .

6. Prove that \sin \theta \cdot \sec \theta=\tan \theta .

7. Prove that \cos \theta \cdot \csc \theta=\cot \theta .

8. If \sin \theta=0.81 , what is \sin (-\theta) ?

9. If \cos \theta=0.5 , what is \cos (-\theta) ?

10. If  \cos \theta=0.25 , what is \sec (-\theta) ?

11. If \csc \theta=0.7 , what is \sin (-\theta) ?

12. How can you tell from a graph if a function is even or odd?

13. Prove \frac{\tan x \cdot \sec x}{\csc x} \cdot \cot x =\tan x .

14. Prove \frac{\sin^2 x \cdot \sec x}{\tan x} \cdot \csc x=1 .

15. Prove \cos x \cdot \tan x =\sin x .

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