Trig identities

Here you will learn about trigonometric identities, including recognizing and working with key trigonometric identities, as well as applying algebraic skills to simplify the identities.

Students first learn how to work with trigonometric identities in Algebra II and expand that knowledge through Precalculus.

What are trigonometric identities?

Trigonometric identities are mathematical equations that involve trigonometric functions, like sine, cosine, and tangent and they are true for all values of the variables involved.

An identity is an equation that is always true.

For example, when expanding the algebraic expression 2(x+1) using the distributive property you get 2 x+2. These two functions of x are equivalent so you can write the statement: 2(x+1) \equiv 2 x+2.

The same is true for trigonometric ratios and expressions.

Let’s take a look at several equivalent expressions that are trigonometric identities.

Using the right triangle:

Where O= Opposite side, A= Adjacent side, and H= Hypotenuse for the right-angled triangle with respect to angle \theta (theta). The following relationships are considered trigonometric identities because they are equivalent, equal or identical.

• \sin(\theta)=\cfrac{O}{H}
• \cos(\theta)=\cfrac{A}{H}
• \tan(\theta)=\cfrac{O}{A}

Where O= Opposite side, A= Adjacent side, and H= Hypotenuse for the right-angled triangle with respect to angle \theta (theta).

Let’s take a look at another trigonometric relationship that is equivalent, equal, or identical, hence making it a trigonometric identity.

This relationship can be proven true using the same right triangle. Substitute \sin \theta with \cfrac{O}{H} and \cos \theta with \cfrac{A}{H}.

This proves that \tan (\theta)=\cfrac{O}{A} which also equals \cfrac{\sin \theta}{\cos \theta} meaning that \tan (\theta)=\cfrac{\sin \theta}{\cos \theta} is always a true relationship so it is a trigonometric identity.

Let’s take a look at another trigonometric relationship that is equivalent, equal, or identical, hence making it a trigonometric identity.

This relationship can be proven true using a right triangle. Notice how values are assigned to each of the sides. Using the numerical values might be easier to see the relationship.

In this case, \sin \theta=\cfrac{3}{5} and \cos \theta=\cfrac{4}{5} so, let’s substitute those values in for \sin \theta and \cos \theta.

This proves that \sin ^2(\theta)+\cos ^2(\theta)=1 is a true relationship meaning that it is equivalent, equal, or identical so it is a trigonometric identity.

\sin ^2(\theta)+\cos ^2(\theta)=1 is known as a Pythagorean theorem identity because you use the Pythagorean theorem to prove it true.

Both, \tan (\theta)=\cfrac{\sin \theta}{\cos \theta} and \sin ^2(\theta)+\cos ^2(\theta)=1 are identities that you can use to prove other identities.

What are trigonometric identities?

Common Core State Standards

How does this apply to high school math?

• High School Functions – Trigonometric Functions (HSF-TF.C.8)
  Prove the Pythagorean identity \sin ^2(\theta)+\cos ^2(\theta)=1 and use it to find \sin(\theta), \, \cos(\theta), or \tan(\theta) given \sin(\theta), \, \cos(\theta), or \tan(\theta) and the quadrant of the angle.

How to prove trigonometric identities

In order to prove trigonometric identities:

1. Choose one side of the identity to simplify.
2. Substitute values using the trigonometric ratios or another trigonometric identity.
3. Simplify the identity.

Trigonometric identities examples

Example 1: the relationship between sin, cos and tan

Prove that the trigonometric identity is true:

1. Choose one side of the identity to simplify.

In this case, use the left hand side of the identity to simplify.

2Substitute values using the trigonometric ratios or another trigonometric identity.

Using the relationships of the sides of a right triangle, you know that:

\sin (\theta)=\cfrac{O}{H}, \, \cos (\theta)=\cfrac{A}{H}, and \tan (\theta)=\cfrac{O}{A}

The equation can be rewritten to be:

3Simplify the identity.

Use the rules of division of fractions:

Both sides of the equation are identical, meaning that the identity is proven to be true.

Example 2: the relationship between sin, cos and tan

Verify the identity:

Example 3: the relationship between sin, cos and tan

Verify the identity:

Example 4: the relationship between sin, cos and tan

Verify the identity:

Example 5: the relationship between sin, cos and tan

Verify the identity:

Example 6: using Pythagoras’ theorem to prove a trigonometric identity

Verify the identity:

Simplify the identity.

\begin{aligned} \left(\cfrac{O}{H}\right)^2+\left(\cfrac{A}{H}\right)^2&=1 \\\\ \cfrac{o^2+A^2}{H^2}&=1 \end{aligned}

From the right triangle and using Pythagorean theorem, you know that A^2+O^2=H^2 .

So, you can substitute A^2+O^2 with H^2 .

\begin{aligned} \cfrac{O^2+A^2}{H^2}&=1 \\\\ \cfrac{H^2}{H^2}&=1 \\\\ 1&=1 \end{aligned}

The identity is verified to be true.

Example 7: proving identities using exact trig values

Verify the identity \cos(60)=1-\sin(30) using the given right triangle.

Substitute values using the trigonometric ratios or another trigonometric identity.

\cos (60)=1-\sin (30)

Using the right triangle and SOHCAHTOA, the following relationship is true.

\sin (30)=\cfrac{O}{H}=\cfrac{1}{2}

Rewrite the identity to be:

\cos (60)=1-\cfrac{1}{2}

Teaching tips for trigonometric identities

• Have students investigate trig identities using inquiry based platforms such as Desmos.

• Instead of having students practice problems with worksheets, have them engage in collaborative learning activities such as jigsaw tasks, scavenger hunts, and/or gallery walks.

• Engage students with team game-playing to review skills with platforms such as Blooket and Quizziz.

Easy mistakes to make

• Treating an identity like an equation
  You do not solve an identity by balancing both sides of the equal sign like an equation. An identity is a statement where two expressions or functions are mathematically identical. For example, 3x+6\equiv{3}(x+2) because when it is factored, the expression becomes 3(x+2).

• Mixing up the right triangle trigonometric ratios
  It is key to remember the three trig ratios that are written as the mnemonic SOHCAHTOA. This way you will not confuse the relationships between the sides and the angles.
  
  \begin{aligned}\sin \theta & =\cfrac{O}{H} \\\\ \cos \theta & =\cfrac{A}{H} \\\\ \tan \theta & =\cfrac{O}{A} \\\\ \csc \theta & =\cfrac{H}{O} \text { (reciprocal of the sine) } \\\\ \sec \theta & =\cfrac{H}{A} \text { (reciprocal of the cosine) } \\\\ \cot \theta & =\cfrac{A}{O} \text { (reciprocal of the tangent) } \end{aligned}

Related trigonometry lessons

• Trigonometry
• Trig functions
• How to find the exact values of a trig function
• Sin cos tan
• Law of sines
• Trig formula for area of a triangle
• Trig formulas
• Law of cosines
• Trig table

The next lessons are

• Circle math
• Sectors, arcs and segments
• Circle theorems

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