Question

The point P on the unit circle that corresponds to a real number t is given. Find \(\sin t, \cos t,\) tan \(t, \csc t, \sec t,\) and cot \(t\). $$ \left(\frac{\sqrt{5}}{3}, \frac{2}{3}\right) $$

Short answer

\( \sin t = \frac{2}{3}, \cos t = \frac{\sqrt{5}}{3}, \tan t = \frac{2 \sqrt{5}}{5}, \csc t = \frac{3}{2}, \sec t = \frac{3 \sqrt{5}}{5}, \cot t = \frac{\sqrt{5}}{2} \)

Step-by-step solution

- Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The coordinates \((x, y)\) on the unit circle are given by \((\cos t, \sin t)\).

- Identify \(\bold{\sin t}\) and \(\bold{\cos t}\)

The coordinates of the point P correspond to \((\cos t, \sin t)\). So, \(\cos t = \frac{\sqrt{5}}{3}\) and \(\sin t = \frac{2}{3}\).

- Calculate \(\bold{\tan t}\)

The tangent of an angle \(t\) is given by \(\tan t = \frac{\sin t}{\cos t} = \frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}\).

- Calculate \(\bold{\csc t}\)

The cosecant of an angle \(t\) is given by \(\csc t = \frac{1}{\sin t} = \frac{1}{\frac{2}{3}} = \frac{3}{2}\).

- Calculate \(\bold{\sec t}\)

The secant of an angle \(t\) is given by \(\sec t = \frac{1}{\cos t} = \frac{1}{\frac{\sqrt{5}}{3}} = \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}\).

- Calculate \(\bold{\cot t}\)

The cotangent of an angle \(t\) is given by \(\cot t = \frac{\cos t}{\sin t} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2}\).

Key concepts

sine and cosine

In trigonometry, the functions sine (sin) and cosine (cos) are fundamental.
When dealing with the unit circle, these functions correspond to the coordinates of any point on the circle.
Specifically, \(\text{cos } t\) gives us the x-coordinate, and \(\text{sin } t\) gives us the y-coordinate.
For the point P given in the exercise, where P is \(\bigg(\frac{\text{\sqrt{5}}}{3}, \frac{2}{3}\bigg)\), we identify that:
\(\text{cos } t = \frac{\text{\sqrt{5}}}{3}\) and \(\text{sin } t = \frac{2}{3}\).Understanding this correspondence helps in determining other trigonometric functions derived from sine and cosine.

tangent

The tangent (tan) of an angle in the unit circle can be found using the sine and cosine values.
It is defined as \(\text{tan } t = \frac{\text{sin } t}{\text{cos } t}\).
For the point P, using \(\text{sin } t = \frac{2}{3}\) and \(\text{cos } t = \frac{\text{\sqrt{5}}}{3}\), we calculate: \(\text{tan } t = \frac{\frac{2}{3}}{\frac{\text{\sqrt{5}}}{3}} = \frac{2}{\text{\sqrt{5}}} = \frac{2 \text{\sqrt{5}}}{5}\).
This form is often used to simplify calculations or for approximation.

cotangent

Cotangent (cot) is the reciprocal of the tangent.
To find cotangent on the unit circle, you use the formula: \(\text{cot } t = \frac{\text{cos } t}{\text{sin } t}\).
Given \(\text{cos } t = \frac{\text{\sqrt{5}}}{3}\) and \(\text{sin } t = \frac{2}{3}\), we find: \(\text{cot } t = \frac{\frac{\text{\sqrt{5}}}{3}}{\frac{2}{3}} = \frac{\text{\sqrt{5}}}{2}\).
Notice that cotangent is always the product of cosine divided by sine
Understanding these relationships can significantly aid in solving trigonometric problems.

secant

Secant (sec) is the reciprocal of cosine.
To calculate it, use the formula: \(\text{sec } t = \frac{1}{\text{cos } t}\).
From the exercise, where \(\text{cos } t = \frac{\text{\sqrt{5}}}{3}\), we get: \(\text{sec } t = \frac{1}{\frac{\text{\sqrt{5}}}{3}} = \frac{3}{\text{\sqrt{5}}} = \frac{3\text{\sqrt{5}}}{5}\).
This means secant evaluates how much the hypotenuse is stretched compared to the base, making it useful in many trigonometric applications.

cosecant

Cosecant (csc) is the reciprocal of sine.
Calculated using: \(\text{csc } t = \frac{1}{\text{sin } t}\), it is essential in many trigonometric expressions.
With \(\text{sin } t = \frac{2}{3}\) from the point P in the exercise, we find: \(\text{csc } t = \frac{1}{\frac{2}{3}} = \frac{3}{2}\).
This cinches the inverse relationship between sine and cosecant.