Exercises: Directed angles and the unit circle

An angle greater than $360º$ represents more than one full turn. Express each of these angles as a number of full turns plus an angle less than $360º$:

• $\alpha = 750º$

• $\beta = 1180º$

• $\gamma = 2000º$

Negative angles are measured clockwise. Express each of these angles as a positive angle less than $360º$ with the same terminal side:

• $\alpha = -60º$

• $\beta = -150º$

• $\gamma = -400º$

State which quadrant the terminal side of each of these angles lies in:

• $\alpha = 95º$

• $\beta = 262º$

• $\gamma = 310º$

• $\delta = -80º$

• $\varepsilon = 500º$

On the unit circle, the sine of an angle is the y-coordinate (ordinate) and the cosine is the x-coordinate (abscissa) of the point where the terminal side intersects the circle, as shown in the figure.

Looking at the figure, calculate with reasoning the value of:

• $\sin 90º$ and $\cos 90º$

• $\sin 180º$ and $\cos 180º$

• $\sin 270º$ and $\cos 270º$

• $\sin 360º$ and $\cos 360º$

Without using a calculator, state the sign (positive or negative) of each of these trigonometric ratios, giving reasons for which quadrant each angle belongs to:

• $\sin 130º$

• $\cos 200º$

• $\tan 320º$

• $\cos 95º$

• $\sin 350º$

• $\tan 140º$

The figure shows an angle $\alpha$ on the unit circle, with the projections of its point $P$ onto the axes.

From the figure, state the sign of $\sin \alpha$, $\cos \alpha$ and $\tan \alpha$, justifying each answer using the coordinates of point $P$.

The figure shows the angle $135º$ on the unit circle, together with its reference triangle.

Given that $\sin 45º = \frac{\sqrt{2}}{2}$ and $\cos 45º = \frac{\sqrt{2}}{2}$, calculate the exact value of $\sin 135º$ and $\cos 135º$, taking into account the quadrant in which point $P$ lies.

Determine which quadrant each of the angles $\alpha$, $\beta$ and $\gamma$ lies in, knowing that:

• $\sin \alpha < 0$ and $\tan \alpha > 0$

• $\cos \beta > 0$ and $\sin \beta < 0$

• $\cos \gamma < 0$ and $\tan \gamma < 0$

Also state the sign of the missing trigonometric ratio in each case.

A Ferris wheel rotates anticlockwise. One of its capsules starts at the lowest point and, after completing a full cycle, has rotated a total of $750º$.

• How many full turns has the capsule made?

• What angle less than $360º$ describes its final position relative to its starting position?

• If the rotation had been $1500º$, where would the capsule end up?

A sailboat sails on a bearing of $220º$. In navigation, bearings are measured clockwise from north; in mathematics, angles are measured anticlockwise from east (the positive $X$ axis).

• Express the bearing of the sailboat as a mathematical angle $\alpha$ between $0º$ and $360º$.

• State which quadrant the sailboat is sailing in and the sign of $\sin \alpha$ and $\cos \alpha$.

Consider the angle $\alpha = 215º$.

• State which quadrant its terminal side lies in.

• Determine the sign of $\sin \alpha$, $\cos \alpha$ and $\tan \alpha$.

• Write $\alpha$ as $180º$ plus an acute angle, and also as a negative angle between $-360º$ and $0º$.

State whether each of the following statements is true or false, giving reasons for your answer:

• There exists an angle $\alpha$ such that $\cos \alpha = 1.4$.

• If $\sin \alpha > 0$, then $\alpha$ belongs to the first quadrant.

• The tangent of $90º$ does not exist.

• $\sin(-40º) = \sin 40º$.

A student wrote the following working in their notebook:

«$\cos 130º = \cos(180º - 50º) = \cos 50º \approx 0.64$»

Find the mistake made, explain why it is a mistake, and calculate the correct value of $\cos 130º$ using a calculator, to four decimal places.

The figure shows two angles, $\alpha$ and $\beta$, on the unit circle whose points on the terminal side are at the same height.

• What is the relationship between the angles $\alpha$ and $\beta$? Express it as an equation.

• If $\sin \alpha = 0.42$ and $\alpha \approx 25º$, find another angle in $[0º, 360º)$ that has the same sine.

• What is the relationship between two angles in $[0º, 360º)$ that have the same cosine? Justify your answer using the unit circle.

Point $P$ in the figure lies on the unit circle, is in the third quadrant, and satisfies $\cos \alpha = -0.8$.

• What is the sign of $\sin \alpha$? And of $\tan \alpha$?

• Calculate the exact value of $\sin \alpha$ using the fundamental identity $\sin^2\alpha + \cos^2\alpha = 1$.

• Calculate $\tan \alpha$.

• Let $\beta = \alpha - 180º$. State which quadrant $\beta$ is in and calculate its three trigonometric ratios.

From the definition $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$:

• Determine all angles $\alpha$ in $[0º, 360º)$ for which $\tan \alpha$ does not exist, justifying your answer.

• Determine all angles $\alpha$ in $[0º, 360º)$ for which $\tan \alpha = 0$.

• Using any integer $k$, write the general expression for all angles for which the tangent does not exist.