Trig Functions

Trigonometric functions connect angles with ratios of lengths and with positions on a circle. The three most familiar are sine, cosine, and tangent. Their reciprocals are cosecant, secant, and cotangent.

Right triangles provide a useful starting point. The unit circle then extends the definitions to angles beyond the acute angles in a right triangle.

Sine, cosine, and tangent in a right triangle

Choose an acute angle \(\theta\). The hypotenuse is opposite the right angle. The opposite side is across from \(\theta\), while the adjacent side is the other side meeting \(\theta\), excluding the hypotenuse.

\[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\]
\[\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\]
\[\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.\]

The mnemonic SOH–CAH–TOA summarizes these three ratios. Opposite and adjacent depend on the angle chosen; switching to the other acute angle switches those two sides.

A worked example

Suppose a right triangle has side lengths 5, 12, and 13. Choose the angle opposite the side of length 5. Its adjacent side has length 12, and the hypotenuse has length 13. Therefore,

\[\sin\theta=\frac5{13},\qquad\cos\theta=\frac{12}{13},\qquad\tan\theta=\frac5{12}.\]

The lengths satisfy \(5^2+12^2=13^2\), as required by the Pythagorean theorem.

To find the angle, use an inverse trigonometric function:

\[\theta=\arctan(5/12)\approx22.62^\circ.\]

Check that your calculator is in degree mode if you want this answer in degrees. In radian mode, the same angle is approximately 0.3948 radians.

The reciprocal functions

The remaining three functions reverse the corresponding ratios:

\[\csc\theta=\frac1{\sin\theta},\qquad\sec\theta=\frac1{\cos\theta},\]
\[\cot\theta=\frac{\cos\theta}{\sin\theta},\]

where the denominators in each expression are nonzero. Wherever tangent is defined and nonzero, cotangent also equals \(1/\tan\theta\). In the triangle above, the three reciprocal functions have values \(13/5\), \(13/12\), and \(12/5\), respectively.

Many scientific calculators provide sin, cos, and tan keys directly. Reciprocal functions can usually be evaluated from those keys; available buttons depend on the calculator.

The unit circle

On a circle of radius 1 centered at the origin, an angle \(\theta\) measured counterclockwise from the positive x-axis reaches the point

\[(x,y)=(\cos\theta,\sin\theta).\]

This definition handles negative angles and angles greater than 90 degrees. Tangent is \(y/x\) wherever \(x\ne0\). For example, at 90°, cosine is zero, sine is 1, and tangent is undefined.

The relationship \(x^2+y^2=1\) gives the identity

\[\sin^2\theta+\cos^2\theta=1.\]

Sine and cosine repeat every full turn: 360° or \(2\pi\) radians. Tangent repeats every 180° or \(\pi\) radians, on its domain.

Degrees, radians, and inverse functions

Because \(180^\circ=\pi\) radians, 30° equals \(\pi/6\) radians. A numerical input such as “30” produces different results in degree and radian modes.

The notation \(\sin^{-1}(x)\) on a calculator normally means \(\arcsin(x)\), an inverse function used to recover a principal angle. It does not mean \(1/\sin(x)\), which is cosecant.

An application: estimating a height

Suppose you stand 18 meters horizontally from the base of a vertical object. From an instrument 1.5 meters above level ground, the angle of elevation to its top is 35°. Under this idealized geometry,

\[h=1.5+18\tan35^\circ\approx14.10\ \mathrm{m}.\]

The 1.5 meters accounts for the instrument’s height. The calculation assumes level ground and a vertical object; measurements with uncertainty produce an approximate result.

Function tangent versus tangent line

The trigonometric function \(\tan\theta\) is a ratio. A tangent line is a geometric object associated with a curve. The terms are related historically, but they are not interchangeable definitions.

Continue with trigonometric ratios and radian measure.