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 \(t\). $$ \left(-\frac{\sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right) $$

Short answer

sin t = \frac{2\sqrt{5}}{5}, cos t = -\frac{\sqrt{5}}{5}, tan t = -2, csc t = \frac{\sqrt{5}}{2}, sec t = -\sqrt{5}, cot t = -\frac{1}{2}

Step-by-step solution

Identify coordinates on the unit circle

The coordinates given are \(-\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\). On the unit circle, these coordinates represent \(\cos t\) and \(\sin t\) respectively.

Find \(\sin t\)

Using the coordinates, \(\sin t = \frac{2\sqrt{5}}{5}\).

Find \(\cos t\)

Using the coordinates, \(\cos t = -\frac{\sqrt{5}}{5}\).

Find \(\tan t\)

Using the identity \(\tan t = \frac{\sin t}{\cos t}\), we get: \[\tan t = \frac{\frac{2\sqrt{5}}{5}}{-\frac{\sqrt{5}}{5}} = -2\]

Find \(\csc t\)

Using the identity \(\csc t = \frac{1}{\sin t}\), we get: \[\csc t = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2}\]

Find \(\sec t\)

Using the identity \(\sec t = \frac{1}{\cos t}\), we get: \[\sec t = \frac{1}{-\frac{\sqrt{5}}{5}} = -\frac{5}{\sqrt{5}} = -\sqrt{5}\]

Find \(\cot t\)

Using the identity \(\cot t = \frac{1}{\tan t}\), we get: \[\cot t = \frac{1}{-2} = -\frac{1}{2}\]

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are fundamental in solving trigonometric equations and simplifying expressions. Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 t + \cos^2 t = 1 \)
• Reciprocal Identities: \( \csc t = \frac{1}{\sin t} \), \( \sec t = \frac{1}{\cos t} \), \( \cot t = \frac{1}{\tan t} \)
• Quotient Identities: \( \tan t = \frac{\sin t}{\cos t} \), \( \cot t = \frac{\cos t}{\sin t} \)

These identities help in finding the values of various trigonometric functions given some initial information.

Sine Function

The sine function, denoted as \( \sin t \), represents the y-coordinate of a point on the unit circle at an angle \( t \). In this exercise, the point \( P = \left(-\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\right) \) is given, where \( \sin t = \frac{2\sqrt{5}}{5} \).

• It ranges from -1 to 1.
• It is periodic with a period of \( 2\pi \).
• It is an odd function, meaning \( \sin(-t) = -\sin(t) \).

Cosine Function

The cosine function, denoted as \( \cos t \), represents the x-coordinate of a point on the unit circle at an angle \( t \). For the point \( P = \left(-\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\right) \), \( \cos t = -\frac{\sqrt{5}}{5} \).

• It ranges from -1 to 1.
• It is periodic with a period of \( 2\pi \).
• It is an even function, meaning \( \cos(-t) = \cos(t) \).

Tangent Function

The tangent function, denoted as \( \tan t \), represents the ratio of the sine and cosine functions, \( \tan t = \frac{\sin t}{\cos t} \). Using the given coordinates, \( \tan t = -2 \).

• It can take any real value (range: all real numbers).
• It is periodic with a period of \( \pi \).
• It is an odd function: \( \tan(-t) = -\tan(t) \).

Secant Function

The secant function, denoted as \( \sec t \), is the reciprocal of the cosine function, \( \sec t = \frac{1}{\cos t} \). For this exercise, \( \sec t = -\sqrt{5} \).

• It is undefined when \( \cos t = 0 \).
• It is periodic with a period of \( 2\pi \).
• It shares the even property with the cosine function: \( \sec(-t) = \sec(t) \).

Cosecant Function

The cosecant function, denoted as \( \csc t \), is the reciprocal of the sine function, \( \csc t = \frac{1}{\sin t} \). In this problem, \( \csc t = \frac{\sqrt{5}}{2} \).

• It is undefined when \( \sin t = 0 \).
• It is periodic with a period of \( 2\pi \).
• It is an odd function like sine: \( \csc(-t) = -\csc(t) \).

Cotangent Function

The cotangent function, denoted as \( \cot t \), is the reciprocal of the tangent function, \( \cot t = \frac{1}{\tan t} \). For the given coordinates, \( \cot t = -\frac{1}{2} \).

• It can take any real value (range: all real numbers).
• It is periodic with a period of \( \pi \).
• It is an odd function: \( \cot(-t) = -\cot(t) \).