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:

• $\alpha = -60º$

• $\beta = -150º$

• $\gamma = -400º$

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?

The Met Éireann weather radar located in Shannon rotates counterclockwise, emitting a scanning beam. The orientation is measured as a goniometric angle where East corresponds to the positive $X$-axis ($0^\circ$) and North to the positive $Y$-axis ($90^\circ$). At a certain moment during monitoring, the antenna records an accumulated rotation of $\alpha = -510^\circ$.

1. a) Reduce the angle $\alpha$ to the first turn by expressing it as a number of complete turns plus a positive angle $\beta \in [0^\circ, 360^\circ)$.
2. b) Toward which geographic quadrant (Northeast, Northwest, Southwest, or Southeast) is the radar beam pointing at that moment?
3. c) If the radar locates the core of a cloud at a normalised distance of radius $1$, determine the Cartesian coordinates $(x, y)$ of the point on the unit circle, expressing them in exact form using the trigonometric ratios of its acute reference angle with the $X$-axis.

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 $\cos 360º$

On the unit circle (radius $1$, centre at the origin), the terminal side of an angle $\alpha$ intersects the circle at a point $P(x, y)$ whose abscissa is $x = -0.8$. Furthermore, it is known that $\tg \alpha > 0$.

1. a) Reasonably determine which quadrant angle $\alpha$ lies in from the signs of its abscissa and its tangent.
2. b) Use the fundamental relation on the unit circle ($x^2 + y^2 = 1$) to deduce the exact value of the ordinate $y$ (that is, $\sen \alpha$) and calculate the value of $\tg \alpha$.
3. c) Knowing that the acute angle in the first quadrant whose cosine is $0.8$ is approximately $36.87^\circ$, calculate:

1. The value of the positive angle $\alpha$ belonging to the first turn ($[0^\circ, 360^\circ)$).

2. A negative angle $\beta$ that has exactly the same terminal side as $\alpha$ after completing a full clockwise turn (between $-720^\circ$ and $-360^\circ$).

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.

The arm of an industrial assembly robot has a normalised length of $1$ metre with its base fixed at the origin $(0,0)$. The end of the arm is oriented by an angle $\theta$ measured counterclockwise from the positive $X$-axis, locating itself at a point $P(x, y)$ on the unit circle. To perform a weld inside a part, the end must be positioned in the third quadrant such that its horizontal distance to the vertical axis is exactly twice its vertical distance to the horizontal axis ($|x| = 2 \times |y|$).

1. a) State the algebraic condition above together with the equation of the unit circle ($x^2 + y^2 = 1$) and obtain the exact values of $\cos \theta$, $\sen \theta$, and $\tg \theta$.
2. b) Knowing that the acute reference angle with the horizontal axis satisfies $\tg \theta_{\text{ref}} = 0.5$ (whose approximate value is $26.57^\circ$), determine the measure of the positive angle $\theta \in [0^\circ, 360^\circ)$ that must be programmed into the robot.
3. c) Due to a transmission error in the servomotors, the control computer records an accumulated total rotation of $-1413.43^\circ$. Prove, by reducing it to the first turn, whether the robotic arm has ended up oriented exactly in the required working position or in a different one.

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

• $\alpha = 95º$

• $\beta = 262º$

• $\gamma = 310º$

• $\delta = -80º$

• $\varepsilon = 500º$

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 $P$.

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:

• $\sin 130º$

• $\cos 200º$

• $\tan 320º$

• $\cos 95º$

• $\sin 350º$

• $\tan 140º$

Consider the directed angle $\alpha = 310^\circ$ on the unit circle.

1. a) Without performing calculations with a calculator, estimate with reasoning which quadrant its terminal side lies in and what signs its abscissa (cosine), its ordinate (sine), and its tangent will have.
2. b) The terminal side of $310^\circ$ is $50^\circ$ away from the positive $X$-axis ($360^\circ - 310^\circ$) and $40^\circ$ away from the negative $Y$-axis ($310^\circ - 270^\circ$). From this relative position on the circle, estimate which of the two absolute values must be greater: $|\sen 310^\circ|$ or $|\cos 310^\circ|$.
3. c) Check your estimate by calculating its acute reference angle with respect to the $X$-axis and deducing which leg of the associated right triangle is longer.

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.

Seán has to calculate the trigonometric ratios of the angle $\alpha = 120^\circ$. In his notebook, he writes the following explanation: "Since the terminal side of $120^\circ$ is $30^\circ$ away from reaching the positive vertical $Y$-axis, its reference angle is $30^\circ$. Therefore, $\sin 120^\circ = \sin 30^\circ = 0.5$ and $\cos 120^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$". Identify the conceptual error Seán made when determining the reference angle, explain why the definition always requires taking the horizontal $X$-axis as the reference, and write the correct values of $\sin 120^\circ$, $\cos 120^\circ$, and $\tan 120^\circ$.

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.

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 $0º$.

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.

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.

• $\sin(-40º) = \sin 40º$.

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.

In class, the problem is posed to find all angles $\alpha$ in the first turn ($0^\circ \le \alpha < 360^\circ$) that satisfy the condition $\sen^2 \alpha = \cos^2 \alpha$. Two students propose different methods to solve it:

• Aoife (Goniometric method): She says that, since $\cos \alpha = x$ and $\sen \alpha = y$ on the unit circle, the condition is equivalent to $y^2 = x^2$, that is, $|y| = |x|$. Therefore, the points sought are the intersections of the unit circle with the angle bisectors of the four quadrants ($y = x$ and $y = -x$).

• Seán (Algebraic method): He proposes dividing both sides by $\cos^2 \alpha$ to obtain $\tg^2 \alpha = 1$, which gives $\tg \alpha = 1$ or $\tg \alpha = -1$.

1. a) Apply Aoife's method and identify the four solution angles, geometrically justifying their symmetry with respect to the axes and their reference angle in the first quadrant.
2. b) Analyse Seán's method: what prior mathematical check is required before dividing the entire expression by $\cos^2 \alpha$, and why in this particular case are we sure not to lose any solution?
3. c) Compare both procedures and justify which of the two provides a greater geometric understanding of the behaviour of the signs of the trigonometric ratios in the four quadrants.