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 \(\frac{\pi}{4}\) 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 in trigonometry?

The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the point where the angle meets the circle has coordinates (cos θ, sin θ). That lets sine and cosine work for every angle, including obtuse and negative ones, not only the acute angles of a right triangle.

What is the unit circle used for?

It is a quick way to find exact trig values and their signs. Reading the point at 30°, (√3/2, 1/2), gives cos 30° = √3/2 and sin 30° = 1/2. The circle also shows why sine and cosine repeat every 2π radians, which quadrants make each function positive, and where identities such as sin²θ + cos²θ = 1 come from.

What is the equation of the unit circle?

The equation is x² + y² = 1, because every point on the circle is exactly 1 unit from the origin. Substituting x = cos θ and y = sin θ gives the Pythagorean identity cos²θ + sin²θ = 1. You can check it at 45°: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1.

How do you memorize the unit circle?

Learn only the first quadrant: 30°, 45° and 60° (π/6, π/4, π/3). Their sines are √1/2, √2/2 and √3/2, and their cosines are the same list in reverse. Every other angle is a reflection of these, so you only change signs by quadrant. The phrase All Students Take Calculus reminds you which functions are positive in quadrants I to IV.

What is tan on the unit circle?

Tangent is the y-coordinate divided by the x-coordinate: tan θ = sin θ/cos θ = y/x. At 45°, both coordinates are √2/2, so tan 45° = 1. At 90° the x-coordinate is 0, so tangent is undefined there. Tangent is positive in quadrants I and III, where x and y have the same sign.

How do you find unit circle coordinates?

Find the reference angle, read its first-quadrant point, then fix the signs for the quadrant. For 150°, the reference angle is 30°, which gives (√3/2, 1/2). In quadrant II the x-coordinate is negative, so the point is (−√3/2, 1/2). That means cos 150° = −√3/2 and sin 150° = 1/2.

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 (resource currently unavailable) covers the unit circle, identities, and applications. Pre-Calculus for Beginners is the natural next step.