Download Fundamental Trigonometric Identities

Fundamental Trigonometric Identities
by CHED on May 07, 2017
lesson duration of 41 minutes
under Precalculus
generated on May 07, 2017 at 04:52 am
Tags: Trigonometry
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Generated: May 07,2017 12:52 PM
Fundamental Trigonometric Identities
( 1 hour and 41 mins )
Written By: CHED on July 4, 2016
Subjects: Precalculus
Tags: Trigonometry
Resources
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Content Standard
Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate
system
Performance Standard
Formulate and solve accurately situational problems involving circular functions
Apply appropriate trigonometric identities in solving situational problems
Formulate and solve accurately situational problems involving appropriate trigonometric functions
Formulate and solve accurately situational problems involving the polar coordinate system
Learning Competencies
Illustrate the different circular functions
Introduction 1 mins
In previous lessons, we have defined trigonometric functions using the unit circle and also investigated the graphs of
the six trigonometric functions. This lesson builds on the understanding of the different trigonometric functions by
discovery, deriving, and working with trigonometric identities.
Domain of an Expression or Equation
Consider the following expressions:
What are the real values of the variable x that make the expressions defined in the set of real numbers?
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thefirst
firstexpression,
expression,every
everyreal
realvalue
valueofof x when
whensubstituted
substitutedtotothe
theexpression
expressionmakes
makesit itdefined
definedininthe
theset
setofofreal
real
InInthe
numbers; that is, the value of the expression is real when x is real.
Inthe
thesecond
secondexpression,
expression,not
notevery
everyreal
realvalue
valueof
ofxxmakes
makesthe
theexpression
expressiondefined
definedin
in R. For example, when x = 0, the
In
expression becomes square root of ?1, which is not a real number.
Here, for square root of (x^2 ? 1) to be defined in R, x must be in (?infinity,?1]U[1,infinity).
In the third expression, the values of x that make the denominator zero make the entire expression undefined.
x^2 ? 3x
3x ? 4 = (x
(x ? 4)(x
4)(x + 1) = 0 <=> x = 4 or x = ?1
Hence, the expression x/(x^2 ? 3x ? 4) is real when x =/= 4 and x =/= ?1.
In the fourth expression, because the expression square root of (x
(x ? 1) is in the denominator, x must be greater than 1.
Although the value of the entire expression is 0 when x = 0, we do not include 0 as allowed value of x because part of
the expression is not real when x = 0.
In the expressions above, the allowed values of the variable x constitute the domain of the expression.
In the expressions above, the domains of the first, second, third, and fourth expressions are R, (?infinity,?1] U
[1,infinity), R \ {?1, 4}, and (1,infinity), respectively.
Example 3.4.1. Determine the domain of the expression/equation.
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Identity and Conditional Equation
Consider the following two groups of equations:
In each equation in Group A, some values of the variable that are in the domain of the equation do not satisfy the
equation (that is, do not make the equation true). On the other hand, in each equation in Group B, every element in the
domain of the equation satisfies the given equation. The equations in Group A are called conditional equations, while
those in Group B are called identities.
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Example 3.4.2. Identify
Identify
whether
given
equation
is an
identity
a conditional equation.
equation. For
For each
each conditional
conditional
whether
thethe
given
equation
is an
identity
or or
a conditional
equation, provide a value of the variable in the domain that does not satisfy the equation.
(1) x^3 ?2 = (x ? cube root of 2) (x^2 + cube root of 2x + cube root of 4)
(2) sin ^2 ? = cos^2 + 1
(3) sin ? = cos ? ? 1
(4) (1 ? square root of x)/(1 + square root of x) = (1 ? (2 square root of x + x))/(1 ? x)
Solution. (1) This is an identity because this is simply factoring of difference of two cubes.
(2) This is a conditional equation. If ? = 0, then the left-hand side of the equation is 0, while the right-hand side is 2.
(3) This is also a conditional equation. If ? = 0, then both sides of the equation are equal to 0. But if ? = pi, then the lefthand side of the equation is 0, while the right-hand side is ?2.
(4) This is an identity because the right-hand side of the equation is obtained by rationalizing the denominator of the
left-hand side.
The Fundamental Trigonometric Identities
Recall that if P(x, y) is the terminal point on the unit circle corresponding to ?, then we have
From the definitions, the following reciprocal and quotient identities immediately follow. Note that these identities hold if
? is taken either as a real number or as an angle.
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We can use these identities to simplify trigonometric expressions.
Example 3.4.3. Simplify:
If P(x, y) is the terminal point on the unit circle corresponding to ?, then x^2 + y^2 = 1. Since sin ? = y and cos ? = x,
we get
sin^2 ? + cos^2 ? = 1.
By dividing both sides of this identity by cos^2 ? and sin^2 ?, respectively, we obtain
tan2^ ? + 1 = sec2^ ? and 1 + cot^2 ? = csc^2 ?.
Teaching Notes
The assumption in the division is that the divisor is nonzero.
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In addition to the eight identities presented above, we also have the following identities.
The first two of the negative identities can be obtained from the graphs of the sine and cosine functions, respectively.
(Please review the discussion on page 145.) The third identity can be derived as follows:
Teaching Notes
The corresponding reciprocal functions follow the same Even-Odd Identities:
csc(??) = ?csc ?
sec(??) = sec?
cot(??) = ?cot ?.
The reciprocal, quotient, Pythagorean, and even-odd identities constitute what we call the fundamental trigonometric
identities.
identities.
We now solve Example 3.2.3 in a different way.
Example 3.4.5. If sin ? = ?3/4 and cos ? > 0. Find cos ?.
Solution. Using the identity sin^2 ? + cos^2 ? = 1 with cos ? > 0, we have
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Example 3.4.6. If sec ? = 5/2 and tan ? < 0, use the identities to find the values of the remaining trigonometric
functions of ?.
Solution. Note that ? lies in QIV.
Proving Trigonometric Identities
We can use the eleven fundamental trigonometric identities to establish other identities. For example, suppose we
want to establish the identity
csc ? ? cot ? = sin ?/(1 + cos ?)
To verify that it is an identity, recall that we need to establish the truth of the equation for all values of the variable in
the domain of the equation. It is not enough to verify its truth for some selected values of the variable. To prove it, we
use the fundamental trigonometric identities and valid algebraic manipulations like performing the fundamental
operations, factoring, canceling, and multiplying the numerator and denominator by the same quantity.
Start on the expression on one side of the proposed identity (preferably the complicated side), use and apply some of
the fundamental trigonometric identities and algebraic manipulations, and arrive at the expression on the other side of
the proposed identity.
Upon arriving at the expression of the other side, the identity has been established. There is no unique technique to
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prove all identities, but familiarity with the different techniques may help.
Example 3.4.7. Prove: sec x ? cos x = sin x tan x.
Solution.
Seatwork 0 mins
Find the domain of the expression/equation.
Seatwork/Homework 3.4.2 0 mins
Seatwork/Homework 3.4.2
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Identify whether the given equation is an identity or a conditional equation. For each conditional equation, provide a
value of the variable in the domain that does not satisfy the equation.
Seatwork/Homework 3.4.3 99 mins
1. Use the identities presented in this lesson to simplify each trigonometric expression.
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2. Given some initial values, use the identities to find the values of the remaining trigonometric functions of ?.
Seatwork/Homework 3.4.4 0 mins
Prove each identity.
1. tan x + cot x = csc x sec x
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2. sec ? + tan ? = 1/(sec ? ? tan ?)
Answer: 2 csc ?
3. sec y + tan y csc y + 1 = tan y
4. 2 csc^2 ? = 1/(1 ? cos ?) + 1/(1 + cos ?)
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Exercise 1 mins
1. Find the domain of the equation.
2. Simplify each expression using the fundamental identities.
3. Given some initial information, use the identities to find the values of the trigonometric functions of ?.
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4. Determine whether the given equation is an identity or a conditional equation. If it is an identity, prove it; otherwise,
provide a value of the variable in the domain that does not satisfy the equation.
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5. Prove the following identities.
6. Express (1 ? sec^2 x)/sec^2 x in terms of sin x.
Answer: ?sin^2 x
7. Express tan x sec x in terms of cos x.
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8. Express all other five trigonometric functions in terms of tan x (allowing ± in the expression).
9. If sec ? ? tan ? = 3, what is sec ? + tan ??
Answer. 1/3
Solution.
tan^2 ? + 1 = sec^2 ?
sec^2 ? ? tan^2 ? = 1
(sec ? ? tan ?)(sec ? + tan ?) = 1
3(sec ? + tan?) = 1
sec ? + tan ? = 1/3
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