10.01 Reciprocal, quotient, and Pythaogrean identities
Equations versus identities
The 'equals' sign $=$= is used to indicate that whatever is on the left of the sign is the same as whatever is on the right.
Its use was introduced by Robert Recorde in 1557. He explained the use of the two parallel lines by saying '... no two things can be more equal'.
When there are algebraic expressions at the two ends of the equals sign, typically with unspecified quantities $x$x and $y$y, it may be that a restricted number of $x,y$x,y pairs exists such that the statement is true. For example, the statement $2x=3y$2x=3y is not true for every pair of numbers $x$x and $y$y, but it is true for special pairs like $(6,4)$(6,4), $(9,6)$(9,6), $(11,7\frac{1}{3})$(11,713) and so on.
The pairs of numbers for which the statement is true are said to be solutions and the statement itself is called an equation.
An equation can have no solutions, exactly one solution or many solutions. There should also be non-solutions.
If it should happen that a statement is true for every possible number or combination of numbers that can be substituted for the unspecified variables, then we call the statement an identity. The sign used for an identity has three parallel lines rather than two, like this: $\equiv$≡. It means that the expressions at the two ends of the sign are equivalent. For example, we could write $\frac{2x}{5}\equiv\frac{4x^2}{10x}$2x5≡4x210x.
An identity cannot be 'solved' in the way an equation can because it is true for all values of the variable - which is not very helpful if one is looking for a particular solution.
Quotient and reciprocal identities
Trigonometric identities are identities that involve trigonometric functions. In fact, you may have been using these identities already based on the definition of each trigonometric function. The following identities hold, provided the denominator of the fraction is not zero.
$\tan\theta\equiv\frac{\sin\theta}{\cos\theta}$tanθ≡sinθcosθ
$\cot\theta\equiv\frac{\cos\theta}{\sin\theta}$cotθ≡cosθsinθ
$\csc\theta\equiv\frac{1}{\sin\theta}$cscθ≡1sinθ
$\sec\theta\equiv\frac{1}{\cos\theta}$secθ≡1cosθ
$\cot\theta\equiv\frac{1}{\tan\theta}$cotθ≡1tanθ
Using the above identities, we can derive the values of all six of the trigonometric functions if we are given the value of one of them. We can also rewrite trig expressions in equivalent but more concise ways.
Practice questions
Question 1
Find $\csc\theta$cscθ if $\sin\theta=\frac{4}{7}$sinθ=47.
Question 2
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Given that $\sin x=\frac{4}{5}$sinx=45 and $\cos x=\frac{3}{5}$cosx=35, find $\tan x$tanx.
Pythagorean identities
The Pythagorean theorem concerning right triangles can be expressed by means of trigonometric ratios. Suppose a right triangle has legs $a$a and $b$b, and hypotenuse $c$c. Let the angle opposite side $a$a be $\theta$θ.
We know that $a^2+b^2=c^2$a2+b2=c2 by Pythagoras. If we now divide the equation through by $c^2$c2, we obtain the form $\frac{a^2}{c^2}+\frac{b^2}{c^2}=1$a2c2+b2c2=1.
The fractions $\frac{a}{c}$ac and $\frac{b}{c}$bc are recognized as $\sin\theta$sinθ and $\cos\theta$cosθ respectively. Therefore, we can replace the fractions by the trigonometric ratios to obtain:
$\sin^2\theta+\cos^2\theta\equiv1$sin2θ+cos2θ≡1
To be convinced that the identity holds for angles of any magnitude, we can inspect the unit circle diagram and apply Pythagoras's theorem to the triangles that are formed. Note that although the values of $\sin\theta$sinθ and $\cos\theta$cosθ may be negative, their squares are not.
Algebraic manipulation of the Pythagorean identity above leads to two other Pythagorean identities, which we have below:
$\sin^2\theta+\cos^2\theta\equiv1$sin2θ+cos2θ≡1
$\tan^2\theta+1\equiv\sec^2\theta$tan2θ+1≡sec2θ
$1+\cot^2\theta\equiv\csc^2\theta$1+cot2θ≡csc2θ
We can use the above Pythagorean identities to find trig function values given certain information.
Practice questions
QUESTION 3
The point $\left(x,\frac{4}{5}\right)$(x,45) lies on the unit circle.
Use the identity $\cos^2\left(s\right)+\sin^2\left(s\right)=1$cos2(s)+sin2(s)=1 to find the value of $x$x if $x<0$x<0.
Question 4
Given that $\sin\theta=\frac{\sqrt{3}}{2}$sinθ=√32 for $90^\circ<\theta<180^\circ$90°<θ<180°, find the value of $\cos\theta$cosθ using the Pythagorean identity.
Simplifying trigonometric expressions
We can use the reciprocal, quotient, and Pythagorean identities to simplify trig expressions and establish further trigonometric identities. Let's review some strategies for success.
Rewrite in terms of sine and cosine
Many trig expressions can be simplified if we first rewrite the expression in terms of sine and cosine. After that, we can use identities to simplify the expression further.
Worked example
Question 5
Simplify the expression $\sec x-\tan x\sin x$secx−tanxsinx.
Think: Rewrite the expression in terms of sine and cosine using the reciprocal and quotient identities.
Do: Perform the substitution and simplify as much as possible.
| $\sec x-\tan x\sin x$secx−tanxsinx | $=$= | $\frac{1}{\cos x}-\frac{\sin x}{\cos x}\sin x$1cosx−sinxcosxsinx |
Rewrite in terms of sine and cosine using the reciprocal and quotient identities |
| $=$= | $\frac{1}{\cos x}-\frac{\sin^2\left(x\right)}{\cos x}$1cosx−sin2(x)cosx |
Multiply. |
|
| $=$= | $\frac{1-\sin^2\left(x\right)}{\cos x}$1−sin2(x)cosx |
Add fractions with a common denominator. |
|
| $=$= | $\frac{\cos^2\left(x\right)}{\cos x}$cos2(x)cosx |
Substitute using the Pythagorean identity. |
|
| $=$= | $\cos x$cosx |
Cancel out common factors. |
Simplify by factoring or multiplying
Sometimes, rewriting or simplification might require factoring or multiplying trigonometric expressions first. Factoring trigonometric expressions is the same process as factoring and multiplying polynomials, and has the same recognizable patterns.
Practice question
Question 6
Simplify $\left(\cos\theta-1\right)\left(\cos\theta+1\right)$(cosθ−1)(cosθ+1).
Simplify by combining fractions
Another strategy is to combine fractions by establishing a common denominator. The rules for adding fractions with trig functions are the same as adding fractions with other real numbers.
Practice question
Question 7
Simplify $\frac{\sin\theta}{1+\cos\theta}+\frac{1+\cos\theta}{\sin\theta}$sinθ1+cosθ+1+cosθsinθ.