The Significance of the Unit Circle in Trigonometric Functions

Significance of Quadrants in Trigonometry

1. All functions are positive
2. Sine and cosecant are positive
3. Tangent and cotangent are positive
4. Cosine and secant are positive.

Coordinates on the Unit Circle

Any angle, corresponds to a specific point on the circumference of the unit circle. The x-coordiante of that point is the cosine value of that angle, and the y-coordinate is the sine value.

For example, at \(30^\circ\) or \(\frac{\pi}{6}\) radians, the point on the unit circle has the x-coordinate of \(\frac{\sqrt{3}}{2}\), which is cosine of \(30^\circ\), and a y-coordinate of \(\frac{1}{2}\), which is sine of \(30^\circ\). Similarly, for \(45^\circ\) or \(\frac{\pi}{6}\) radians, sine and cosine are equal \(\frac{\sqrt{2}}{2}\)

Extending to Other Trigonometric Functions

Now that we know how to find the values for sine and cosine of an angle, we can use them to find the values for other trigonometric functions. \(\frac{\sin x}{\cos x} = \tan x\), and the inverse of tangent, is cotangent.

Secant is the inverse of cosine \((\frac{1}{\cos x})\) and cosecant is the inverse of sine \((\frac{1}{\sin x})\)

For example, for the \(\frac{\pi}{4}\) angle, we have:

\(\tan \left(\frac{\pi}{4}\right) = \frac{\sin \left(\frac{\pi}{4}\right)}{\cos \left(\frac{\pi}{4}\right)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1 = \cot \left(\frac{\pi}{4}\right)\)

\(\sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\)

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Symmetry in Trigonometric Functions

• An even function is symmetric around the y-axis, meaning \(f(x)=f(-x)\), for all \(x\).

• An odd function is symmetric with respect to the origin, satisfying \(f(x)=-f(-x)\), for all \(x\).

For example, \(\sin x\) is odd, while \(\cos x\) is even. This can be helpful when we want to convert trigonometric functions to each other. For example, \(\sin 120^\circ = +0.86\), while \(\sin (-120^\circ) = -0.86\). and from unit circle, we know that \(-120^\circ = 240^\circ\), since \(0\) and \(360\) are basically in the same place, so: \(360^\circ – 120^\circ = 240^\circ\)

\(\begin{align*} \sin^2 \left(\frac{\pi}{4}\right) + \cos^2 \left(\frac{\pi}{4}\right) &= \left( \frac{\sqrt{2}}{2} \right)^2 + \left( \frac{\sqrt{2}}{2} \right)^2 \\&= \frac{2}{4} + \frac{2}{4} \\&= \frac{1}{2} + \frac{1}{2} \\&= 1\end{align*}\)

Trigonometric Identities and the Unit Circle

There are various trigonometric identities that help to convert angles of trigonometric functions to other angles, trigonometric functions to one another, eliminate/create exponents, and so on.

These identities can mix up with each other to generate new identities. Some of them are used frequently in trigonometry, while others not as often. For instance, the identity \(\sin^2 x + \cos^2 x = 1\)

Is used very often, and many other useful identities are derived from it. Let’s test it for \(45^\circ\):

Here are a list of useful trigonometric identities:

\(\sin 2\theta = 2 \sin \theta \cos \theta\)

\(\cos 2\theta = 1 – 2\sin^2 \theta = 2\cos^2 \theta – 1 = \cos^2 \theta – \sin^2 \theta = \frac{\cot^2 \theta – 1}{2\cot \theta}\)

\(\tan 2\theta = \frac{2 \tan \theta}{1 – \tan^2 \theta}\)

\(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\)

\(\sin^2 \theta = \frac{1 – \cos 2\theta}{2}\)

\(\sin(a + b) = \sin a \cdot \cos b + \sin b \cdot \cos a\)

\(\sin(a – b) = \sin a \cdot \cos b – \sin b \cdot \cos a\)

\(\cos(a + b) = \cos a \cdot \cos b – \sin a \cdot \sin b\)

\(\cos(a – b) = \cos a \cdot \cos b + \sin a \cdot \sin b\)

\(\tan(a + b) = \frac{\tan a + \tan b}{1 – \tan a \cdot \tan b}\)

\(\tan(a – b) = \frac{\tan a – \tan b}{1 + \tan a \cdot \tan b}\)

\(\cot(a + b) = \frac{\cot a \cdot \cot b – 1}{\cot b + \cot a}\)

\(\cot(a – b) = \frac{\cot a \cdot \cot b + 1}{\cot b – \cot a}\)

\(\sin a + \sin b = 2 \sin \left(\frac{a + b}{2}\right) \cdot \cos \left(\frac{a – b}{2}\right)\)

\(\sin a – \sin b = 2 \cos \left(\frac{a + b}{2}\right) \cdot \sin \left(\frac{a – b}{2}\right)\)

\(\cos a + \cos b = 2 \cos \left(\frac{a + b}{2}\right) \cdot \cos \left(\frac{a – b}{2}\right)\)

\(\cos a – \cos b = -2 \sin \left(\frac{a + b}{2}\right) \cdot \sin \left(\frac{a – b}{2}\right)\)

\(\tan x + \tan y = \frac{\sin(x + y)}{\cos x \cdot \cos y}\)

\(\tan x – \tan y = \frac{\sin(x – y)}{\cos x \cdot \cos y}\)

\(\cot x + \cot y = \frac{\sin(x + y)}{\sin x \cdot \sin y}\)

\(\cot x – \cot y = \frac{\sin(y – x)}{\sin x \cdot \sin y}\)

\(\sin a \cdot \cos b = \frac{1}{2}[\sin(a + b) + \sin(a – b)]\)

\(\cos a \cdot \cos b = \frac{1}{2}[\cos(a + b) + \cos(a – b)]\)

\(\sin a \cdot \sin b = -\frac{1}{2}[\cos(a + b) – \cos(a – b)]\)

\(\sin 3a = -4 \sin^3 a + 3 \sin a\)

\(\cos 3a = 4 \cos^3 a – 3 \cos a\)

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Frequently Asked Questions

What is the unit circle?

A circle of radius 1 centered at the origin of the coordinate plane. Every point on the circle satisfies x^2 + y^2 = 1.

Why is the unit circle important for trigonometry?

Because it gives a clean geometric definition of sine and cosine that works for ALL angles — not just the acute angles of a right triangle. For any angle theta, the point on the unit circle at angle theta from the positive x-axis is (cos(theta), sin(theta)).

How are sine and cosine defined on the unit circle?

For an angle theta measured counterclockwise from the positive x-axis, cos(theta) is the x-coordinate of the corresponding point on the circle, and sin(theta) is the y-coordinate.

What is the Pythagorean identity?

sin^2(theta) + cos^2(theta) = 1. It follows directly from x^2 + y^2 = 1 for any point on the unit circle.

How is tan(theta) defined on the unit circle?

tan(theta) = sin(theta)/cos(theta) = y/x, where (x, y) is the unit-circle point at angle theta. Undefined where x = 0 (at theta = pi/2, 3pi/2, etc.).

What are the standard reference angles?

pi/6 (30 deg), pi/4 (45 deg), pi/3 (60 deg), and their multiples. The unit circle values for these angles are worth memorizing.

What is cos(60 degrees)?

1/2. The point on the unit circle at 60° is (1/2, sqrt(3)/2), so cos = 1/2 and sin = sqrt(3)/2.

Why are sine and cosine periodic?

Because going around the unit circle one full revolution (2*pi radians, or 360°) brings you back to the same point. So sin(theta + 2 pi) = sin(theta) for any theta.

What are radians vs degrees?

Radians measure angles by arc length on the unit circle. One full revolution = 2 pi radians = 360 degrees. So pi radians = 180°, pi/2 radians = 90°, and so on.

How does the unit circle help in calculus?

Calculus formalizes the rates of change of sine and cosine, and the unit-circle picture makes the derivative formulas (d/dx sin x = cos x, d/dx cos x = -sin x) geometrically obvious.

Related Lessons You May Like

• How to use right-triangle trigonometry
• How to solve the ambiguous case (SSA) in trigonometry
• How to find the slope of a line
• How to write equation of parallel and perpendicular lines
• How to find the axis of symmetry of quadratic functions

For a workbook on trigonometry, Trigonometry for Beginners covers the unit circle, identities, and applications. Pre-Calculus for Beginners is the natural next step.