Tangent Identities

Apply double angle formulas in trig expressions and equations. Math 30-1.

Lesson 5.5 of Trigonometric Identities in Math 30-1: Alberta curriculum lessons.

What you'll learn in this lesson

• The Tangent Identities & Finding Exact Values. Learn the sum, difference, and double angle identities for tangent, practice collapsing expressions into a single trig ratio, and use the identities to find exact values of non-special angles.
• Application & Solving Equations. Apply tangent identities when given angle information in quadrants, find exact values from partial trig ratios, and solve equations by recognizing and collapsing tangent identity patterns.

Inside the lesson: a free preview

Where Do They Come From?

Similar to sine and cosine, the sum or difference and double angle identities for tangent can be obtained from the sin and cos identities.

Sum and Difference Identities for Tangent

Sum Identity. (α + β) = (α + β)/(1 - αβ)

Difference Identity. (α - β) = (α - β)/(1 + αβ)

Double Angle Identity. 2α = (2α)/(1 - ^2α)

📌 All of these identities are on your formula sheet.

Writing as a Single Trig Ratio

Always look to see if an expression matches the pattern of one of the tangent identities, if it does, you can collapse it into a single ratio immediately.

Finding Exact Values

The tangent identities can be used to find exact values by breaking angles down into two angles whose exact trig values you already know.

📌 Remember: exact value means NO decimals!

Given Angle Information

Sometimes you won't be given the angle directly. You'll need to find missing trig ratios using the quadrant information before applying the identity.

Solving Equations Using Tangent Identities

Sometimes a trig equation will contain an expression that matches a tangent identity. Simplify using the identity first, then solve the resulting equation.

Worked example: Write each expression as a single trig ratio, then evaluate where possible

Part a) ((π)/(3) - (π)/(6))/(1 + (π)/(3)(π)/(6)). Formula: (α - β) = (α - β)/(1 + αβ)

With α = (π)/(3) and β = (π)/(6), the expression matches the difference identity exactly:

= ((π)/(3) - (π)/(6)) = ((2π)/(6) - (π)/(6)) = (π)/(6)

= (√(3))/(3)

Part b) ((7x) + (4x))/(1 - (7x)(4x)). Formula: (α + β) = (α + β)/(1 - αβ)

With α = 7x and β = 4x, the expression matches the sum identity exactly:

= (7x + 4x)

= 11x

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