Tangent

High School

Definition

A trigonometry function written as Tan or as Tan θ. Tanθ = Sin θ/Cos θ. In a right angle triangle, the tangent of angle θ is the ratio of the opposite side over the adjacent side of the angle θ. In plane geometry, a perpendicular line to the circles radius, that touches a circle at just one point but does not pass through is a tangent.

Worked examples

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

Tangent is sine divided by cosine, so we can compute it from the unit circle values.

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

In a right triangle with opposite side 3 and adjacent side 4, tangent of the angle is 3/4.

A line touches the circle at one point perpendicular to the radius there.

In geometry, a tangent line meets the circle at exactly one point without crossing through it.

Common mistakes

\(\tan(\theta) = \frac{\)adjacent\(}{\)opposite\(}\) → \(\tan(\theta) = \frac{\)opposite\(}{\)adjacent\(}\) Students often flip the ratio; remember tangent puts opposite on top.
\(\tan(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) → \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) Tangent is sine over cosine, not the reverse.
A tangent line can cross through the circle at two points. → A tangent touches the circle at exactly one point. A line intersecting at two points is a secant, not a tangent.

Where you'll use it next

You'll use tangent to solve right-triangle problems in trigonometry, graph tan functions and analyze periodic behavior in precalculus, work with derivatives and slopes in calculus, and apply tangent lines in curve analysis and optimization.

Found in 1 StudyPug lesson

Mastering Tangent Properties in Circles: From Basics to Advanced Theorems

Grade 9 Math

Dive into the world of tangent properties in circles. Learn key theorems, solve complex problems, and discover real-world applications. Boost your geometry skills with our comprehensive guide.

9th Grade9th