Equivalent Expressions
Equivalent Trigonometric Expressions are expressions that contain the same value for all values of the variable. Equivalent expressions can be verified by using graphing technology. If both expressions overlap over the same domain, this means that the expressions are equivalent.
There are a couple of techniques you can use in order to derive equivalent trigonometric expressions:
- Use the Right Triangle: You can use a right triangle to derive equivalent trigonometric expressions that form the cofunction identities, such as \(\sin(x) = \cos \left(\cfrac{\pi}{2} - x\right)\)
- Use the Unit Circle: You can use the Unit Circle along with transformations to derive equivalent trigonometric expressions that form other trigonometric identities such as \(\cos(x) + \cfrac{\pi}{2} = -\sin(x)\)
Cofunction Identities
Cofunction Identities are trigonometric identities that show the relationship of complementary angles and trigonometric functions/ratios between sine and cosine, tangent and cotangent, and secant and cosecant. They are also referred to as corelated angle identities.
Outlined below is a table containing all possible cofunction identities featuring \(\pi⁄2\):
| \(\sin(x) = \cos\left(\cfrac{\pi}{2} - x\right)\) | \(\cos(x) = \sin\left(\cfrac{\pi}{2} - x\right)\) | \(\sin(x + \cfrac{\pi}{2}) = \cos(x)\) | \(\cos(x + \cfrac{\pi}{2}) = -\sin(x)\) |
| \(\tan(x) = \cot\left(\cfrac{\pi}{2} - x\right)\) | \(\cot(x) = \tan\left(\cfrac{\pi}{2} - x\right)\) | \(\tan(x + \cfrac{\pi}{2}) = -\cot(x)\) | \(\cot(x + \cfrac{\pi}{2}) = -\tan(x)\) |
| \(\csc(x) = \sec\left(\cfrac{\pi}{2} - x\right)\) | \(\sec(x) = \csc\left(\cfrac{\pi}{2} - x\right)\) | \(\csc(x + \cfrac{\pi}{2}) = \sec(x)\) | \(\sec(x + \cfrac{\pi}{2}) = -\csc(x)\) |
As you can see, all of the identities are split into 3 pairs: sine and cosine, tangent and cotangent, and secant and cosecant. All of these identities involve either adding \(\pi/2\) to \(x\) or subtracting \(x\) from \(\pi/2\).
Example
Given that \(\sin\left(\cfrac{\pi}{3}\right) = \cfrac{\sqrt{3}}{2}\), use an equivalent trigonometric function to show that \(\cos\left(\cfrac{\pi}{6}\right) = \cfrac{\sqrt{3}}{2}\).
First, we can draw a Special Triangle in order to derive the equivalent trigonometric expressions:
Using this triangle, we can derive the trigonometric expression for \(\sin\left(\cfrac{\pi}{3}\right)\) as \(\cfrac{\sqrt{3}}{2}\). Likewise, we can derive the trigonometric expression for \(\cos\left(\cfrac{\pi}{6}\right)\) as such:
\(\cos\left(\cfrac{\pi}{6}\right) = \cfrac{\sqrt{3}}{2}\)
\(\cfrac{\sqrt{3}}{2} = \cfrac{\sqrt{3}}{2}\)
We can verify our results algebraically by using the applicable cofunction identify:
\(\sin\left(\cfrac{\pi}{3}\right) = \cos\left(\cfrac{\pi}{2} - \cfrac{\pi}{3}\right)\)
\(\sin\left(\cfrac{\pi}{3}\right) = \cos\left(\cfrac{3\pi}{6} - \cfrac{2\pi}{6}\right)\)
\(\sin\left(\cfrac{\pi}{3}\right) = \cos\left(\cfrac{1\pi}{6}\right)\)
\(\cfrac{\sqrt{3}}{2} = \cfrac{\sqrt{3}}{2}\)
LS \(=\) RS
Upon performing calculations, we can verify that both sides are equal to each other.
First, we can draw a Unit Circle with center \(O\). We can then draw the terminal arm for an angle \(x\) in the first quadrant, \(x\in[0, \pi/2]\). Let the intersection of the Unit Circle and the terminal arm of angle 𝑥 be represented by point \(P\):
We can transform \(P\) to \(P’\) by applying a rotation of \(3𝜋/2\) counterclockwise about the origin. From this a triangle, \(∆𝑃′𝑄′𝑂\), can be formed by drawing a vertical line from point \(P’\) to point \(Q’\) on the 𝑥-axis:
We can verify our results algebraically by using the applicable cofunction identify:
\(\sin\left(\cfrac{\pi}{4} + \cfrac{\pi}{2}\right) = \cos\left(\cfrac{\pi}{4}\right)\)
\(\sin\left(\cfrac{\pi}{4} + \cfrac{2\pi}{4}\right) = \cos\left(\cfrac{\pi}{4}\right)\)
\(\sin\left(\cfrac{3\pi}{4}\right) = \cos\left(\cfrac{\pi}{4}\right)\)
\(\cfrac{\sqrt{2}}{2} = \cfrac{\sqrt{2}}{2}\)
LS \(=\) RS
Upon performing calculations, we can verify that both sides are equal to each other.
Compound Angles
A Compound Angle is the algebraic sum of \(2\) or more angles. They can be represented using Trigonometric identities.
You can develop Compound Angle Formulas by using algebra and the Unit Circle. Once you have developed one compound angle formula, you can develop others by applying equivalent trigonometric expressions.
The compound angle, or addition and subtraction, formulas for sine and cosine are:
- \(\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\)
- \(\sin(x-y) = \sin(x)\cos(y) - \cos(x)\sin(y)\)
- \(\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)\)
- \(\cos(x-y) = \cos(x)\cos(y) + \sin(x)\sin(y)\)
Example
Use an appropriate compound angle formula to express the product of \(\sin\cfrac{3\pi}{2}\cos\cfrac{\pi}{6} + \cos\cfrac{3\pi}{2}\sin\cfrac{\pi}{6}\) as a single trigonometric function, and then determine its exact value.
First, we can write the expression as a single trigonometric function by converting it using the appropriate compound angle formula:
\(\sin\left(\cfrac{3\pi}{2} + \cfrac{\pi}{6}\right)\)
Next, we can evaluate the function to determine its exact value. We can do so by adding its values under a common denominator and then diving the entire value by the largest common factor.
\(\sin\left(\cfrac{5\pi}{3}\right)\)
\(-\cfrac{\sqrt{3}}{2}\)
Upon evaluating the function, we can determine its exact value as \(\boldsymbol{-\cfrac{\sqrt{3}}{2}}\).
First, we can expand the expression by using the corresponding compound angle formula:
\(= \cos\left(\cfrac{7\pi}{4}\right)\cos\left(\cfrac{2\pi}{3}\right) + \sin\left(\cfrac{7\pi}{4}\right)\sin\left(\cfrac{2\pi}{3}\right)\)
Next, we can evaluate each respective trigonometric function to determine their exact values. We can do so by using Special Angles.
Then, we can simplify the expression by expanding it and collecting terms:
\(= \cfrac{\sqrt{2} - \sqrt{6}}{4}\)
After, we can remove a common factor from the numerator, \(\sqrt{2}\). We can also rewrite the denominator as \(\sqrt{2}(2\sqrt{2})\). We can then simplify further by cancelling like terms on both sides:
\(= \cfrac{1 - \sqrt{3}}{2\sqrt{2}}\)
Upon evaluating the function, we can determine its exact value as \(\boldsymbol{\cfrac{1 - \sqrt{3}}{2\sqrt{2}}}\).