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Basic Trigonometric Values

Some basic values of each trigonometric function can be found by analyzing the symmetries present in the unit circle for various special arc lengths. These arc lengths are equivalent to degree measures of $0^\circ, 30^\circ, 45^\circ, 60^\circ$,   and $90^\circ$, and their multiples.

Trig Values of Multiples of $\dfrac{\pi}{2}$

$\theta$ $\sin\theta$ $\cos\theta$ $\tan\theta$ $\cot\theta$ $\sec\theta$ $\csc\theta$
0 0 1 0 undefined 0 undefined
$\dfrac{\pi}{2}$ 1 0 undefined 0 undefined 0
$\pi$ 0 $-1$ 0 undefined $-1$ undefined
$\dfrac{3\pi}{2}$ $-1$ 0 undefined 0 undefined $-1$

Proof: Since the circumference of a circle is given by the formula   $C=2\pi r$,   the circumference of the unit circle is $2\pi$. One-fourth of this distance, $\dfrac{\pi}{2}$, will be the arc length of the circle between adjacent coordinate axes. Therefore, the point with coordinates   $(0,1)$   will correspond to the arc length $\dfrac{\pi}{2}$, and the Unit Circle Definition will provide the values of the six trigonometric functions of $\dfrac{\pi}{2}$. The proofs for the other arc lengths are similar.♦

Stated in terms of degrees, the preceding results would be the trigonometric values for $0^\circ, 90^\circ, 180^\circ$, and $270^\circ$. Because none of these values is an acute angle, there is no proof involving triangles for these values.

Trig Values of Multiples of $\dfrac{\pi}{4}$

$\theta$ $\sin\theta$ $\cos\theta$ $\tan\theta$ $\cot\theta$ $\sec\theta$ $\csc\theta$
$\dfrac{\pi}{4}$ $\dfrac{\sqrt{2}}{2}$ $\dfrac{\sqrt{2}}{2}$ 1 1 $\sqrt{2}$ $\sqrt{2}$

Proof: Since the line   $y=x$   bisects the first quadrant, it intersects the unit circle at an arc length of $\dfrac{\pi}{4}$ from the point   $(1,0)$.   Substituting this equation into the equation of the unit circle, we have   $x^2+x^2=1$,   which can be solved to obtain   $x=\dfrac{\sqrt{2}}{2}$.  From the substitution, the variable $y$ has the same value. The ratio and reciprocal identities can then be used to produce the other four trig values.♦

Alternate Proof: Since an unit circle arc length of $\dfrac{\pi}{4}$ occurs with a $45^\circ$ angle, we consider triangle right triangle $ABC$, with   $C=90^\circ$,   and   $A=B=45^\circ$.  Suppose side $AC$ has length 1. Then side $BC$ also has length 1, and by the Pythagorean Theorem, side $AC$ has length $\sqrt{2}$. Values for the six trigonometric functions follow by the Triangle Ratios Theorem.♦

Trig Values of Multiples of $\dfrac{\pi}{6}$

$\theta$ $\sin\theta$ $\cos\theta$ $\tan\theta$ $\cot\theta$ $\sec\theta$ $\csc\theta$
$\dfrac{\pi}{6}$ $\dfrac12$ $\dfrac{\sqrt{3}}{2}$ $\dfrac{\sqrt{3}}{3}$ $\sqrt{3}$ $\dfrac{2\sqrt{3}}{3}$ 2
$\dfrac{\pi}{3}$ $\dfrac{\sqrt{3}}{2}$ $\dfrac12$ $\sqrt{3}$ $\dfrac{\sqrt{3}}{3}$ 2 $\dfrac{2\sqrt{3}}{3}$

Proof: Let $A$ be the point   $(1,0)$   on the unit circle, and choose points $B$ and $C$ in the first quadrant so that arc length $AB$ is   $\dfrac{\pi}{6}$, and arc length $AC$ is $\dfrac{\pi}{3}$. Assume the coordinates of $B$ are   $(x,y)$,   then by symmetry across the line   $y=x$,   the coordinates of $C$ are   $(y,x)$.   Also, the line segments $AB$ and $BC$ are equal. Applying the distance formula, we obtain the equation   $(x-1)^2+y^2=(y-x)^2+(x-y)^2$.   Since point $B$ lies on the unit circle, we also have   $x^2+y^2=1$.   Solving this system gives  \ $\left(\dfrac{\sqrt{3}}{2},\dfrac12\right)$   as the coordinates of point $B$. The definition and basic identities then produce the values of the six trigonometric functions.♦

Alternate Proof: Let $ABC$ be an equilateral triangle. Let point $D$ be the intersection of side $BC$ with the altitude from vertex $A$. By the Hypotenuse-Leg Theorem of geometry, triangles $ABD$ and $ACD$ are congruent. Therefore the altitude from $A$ bisected side $BC$. If side $AB$ has length 2, then $BD$ has length 1, and by the Pythagorean Theorem, side $AD$ has length $\sqrt{3}$. The six trigonometric functions will follow by the Triangle Ratios Theorem.♦

In addition to these basic values, we can also determine the exact values of the sine function at angles that are multiples of 15 degrees, and multiples of 18 degrees.