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Verifying an identity using algebra and even/odd identities

Verify the identity:

sin 2 ( θ ) cos 2 ( θ ) sin ( θ ) cos ( θ ) = cos θ sin θ

Let’s start with the left side and simplify:

sin 2 ( θ ) cos 2 ( θ ) sin ( θ ) cos ( θ ) = [ sin ( θ ) ] 2 [ cos ( θ ) ] 2 sin ( θ ) cos ( θ ) = ( sin θ ) 2 ( cos θ ) 2 sin θ cos θ sin ( x ) = sin x and cos ( x ) = cos x = ( sin θ ) 2 ( cos θ ) 2 sin θ cos θ Difference of squares = ( sin θ cos θ ) ( sin θ + cos θ ) ( sin θ + cos θ ) = ( sin θ cos θ ) ( sin θ + cos θ ) ( sin θ + cos θ ) = cos θ sin θ
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Verify the identity sin 2 θ 1 tan θ sin θ tan θ = sin θ + 1 tan θ .

sin 2 θ 1 tan θ sin θ tan θ = ( sin θ + 1 ) ( sin θ 1 ) tan θ ( sin θ 1 ) = sin θ + 1 tan θ

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Verifying an identity involving cosines and cotangents

Verify the identity: ( 1 cos 2 x ) ( 1 + cot 2 x ) = 1.

We will work on the left side of the equation.

( 1 cos 2 x ) ( 1 + cot 2 x ) = ( 1 cos 2 x ) ( 1 + cos 2 x sin 2 x ) = ( 1 cos 2 x ) ( sin 2 x sin 2 x + cos 2 x sin 2 x ) Find the common denominator . = ( 1 cos 2 x ) ( sin 2 x + cos 2 x sin 2 x ) = ( sin 2 x ) ( 1 sin 2 x ) = 1
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Using algebra to simplify trigonometric expressions

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

For example, the equation ( sin x + 1 ) ( sin x 1 ) = 0 resembles the equation ( x + 1 ) ( x 1 ) = 0 , which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.

Another example is the difference of squares formula, a 2 b 2 = ( a b ) ( a + b ) , which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.

Writing the trigonometric expression as an algebraic expression

Write the following trigonometric expression as an algebraic expression: 2 cos 2 θ + cos θ 1.

Notice that the pattern displayed has the same form as a standard quadratic expression, a x 2 + b x + c . Letting cos θ = x , we can rewrite the expression as follows:

2 x 2 + x 1

This expression can be factored as ( 2 x + 1 ) ( x 1 ) . If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for x . At this point, we would replace x with cos θ and solve for θ .

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Rewriting a trigonometric expression using the difference of squares

Rewrite the trigonometric expression using the difference of squares: 4 cos 2 θ 1.

Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1. This is the difference of squares.

4 cos 2 θ 1 = ( 2 cos θ ) 2 1 = ( 2 cos θ 1 ) ( 2 cos θ + 1 )
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Rewrite the trigonometric expression using the difference of squares: 25 9 sin 2 θ .

This is a difference of squares formula: 25 9 sin 2 θ = ( 5 3 sin θ ) ( 5 + 3 sin θ ) .

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Questions & Answers

f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE
Thanks for this helpfull app
Axmed Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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