Tangent (in Trigonometry)

High School

Definition

Known as tan, and usually written as tan θ¸ in trigonometry. Tangent can be represented as sine and cosine, since tan θ = sin θ/cos θ. Simplifying down sin θ¸ divided by cos θ, you will be left with tan = \(opposite \over adjacent\). The memory device SOHCAHTOA can help you to remember this.

Worked examples

\(\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\)

Tangent equals sine divided by cosine; simplify the fraction to find the ratio.

\(\tan \theta = \frac{\)opposite\(}{\)adjacent\(} = \frac{3}{4}\)

In a right triangle, tangent is the ratio of the side opposite the angle to the side adjacent to it.

Common mistakes

\(\tan \theta = \frac{\)adjacent\(}{\)opposite\(}\) → \(\tan \theta = \frac{\)opposite\(}{\)adjacent\(}\) Students reverse the ratio; remember SOHCAHTOA — tangent is opposite over adjacent.
\(\tan \theta = \frac{\cos \theta}{\sin \theta}\) → \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) Tangent is sine divided by cosine, not the other way around.
\(\tan 90^\circ = 1\) → \(\tan 90^\circ\) is undefined At 90°, cosine is zero, so dividing sine by cosine is undefined.

Where you'll use it next

You'll use tangent to solve right-triangle problems in trigonometry, analyze periodic functions and their graphs, work with identities and equations, and model slopes and angles in calculus and physics.

Found in 1 StudyPug lesson

Mastering the Tangent Ratio in Trigonometry

Precalculus

Unlock the power of the tangent ratio to calculate angles and sides in right triangles. Enhance your problem-solving skills and excel in trigonometry with our comprehensive guide.

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