Oriented Angles and the Unit Circle
Until now, sine, cosine, and tangent only made sense for acute angles, the ones that fit inside a right triangle. But a GPS needs to calculate the angle of a $200º$ curve, a Ferris wheel keeps turning through $750º$ without stopping, and the blade of a wind turbine passes through the top, the bottom, and the sides. We need a definition that works for any angle, no matter how large, small, or negative. That is the unit circle (also called the goniometric circle).
Angles of More Than One Turn, and Negative Angles
An angle does not have to stay between $0º$ and $360º$. If something keeps rotating, the angle it sweeps out can be $750º$, $2000º$, or more: every $360º$ is one complete turn. By convention, angles are measured from the positive $X$-axis, and are positive if the rotation is counterclockwise and negative if it is clockwise.
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 $360º$: $\alpha = 750º$…
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 $360º$:
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$\alpha = 750º$
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$\beta = 1180º$
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$\gamma = 2000º$
Negative angles are measured clockwise. Express each of these angles as a positive angle less than $360º$ with the same terminal side: $\alpha = -60º$ $\beta = -150º$…
Negative angles are measured clockwise. Express each of these angles as a positive angle less than $360º$ with the same terminal side:
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$\alpha = -60º$
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$\beta = -150º$
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$\gamma = -400º$
The Unit Circle: Sine and Cosine as Coordinates
Draw a circle of radius $1$ centered at the origin of a Cartesian plane. For any angle $\alpha$ measured from the positive $X$-axis, its terminal side meets that circle at exactly one point $P$. That is the key idea: the cosine of $\alpha$ is the $x$-coordinate of $P$, and the sine of $\alpha$ is its $y$-coordinate.
For an acute angle, this definition coincides exactly with the right-triangle one: the radius of the circle (the hypotenuse) measures $1$, so the opposite leg divided by the hypotenuse is simply the $y$-coordinate of $P$.
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…
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:
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$\sin 90º$ and $\cos 90º$
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$\sin 180º$ and $\cos 180º$
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$\sin 270º$ and $\cos 270º$
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$\sin 360º$ and $\cos 360º$
The Four Quadrants and the Sign of the Ratios
The axes divide the unit circle into four quadrants, numbered counterclockwise starting from the positive $X$-axis. The quadrant in which the terminal side of $\alpha$ lies determines, without calculating anything, whether $\sin\ \alpha$ and $\cos\ \alpha$ are positive or negative: you just look at the sign of the $y$- and $x$-coordinates in that region.
State which quadrant the terminal side of each of these angles lies in: $\alpha = 95º$ $\beta = 262º$ $\gamma = 310º$ $\delta = -80º$ $\varepsilon = 500º$
State which quadrant the terminal side of each of these angles lies in:
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$\alpha = 95º$
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$\beta = 262º$
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$\gamma = 310º$
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$\delta = -80º$
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$\varepsilon = 500º$
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:…
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:
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$\sin 130º$
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$\cos 200º$
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$\tan 320º$
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$\cos 95º$
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$\sin 350º$
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$\tan 140º$
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 $\sin \alpha$,…
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 $P$.
Angles Larger Than 90º: The Reference Triangle
How do you calculate, for example, $\sin\ 135º$ without a calculator? The key is the reference triangle: the right triangle formed by the terminal side, its projection onto the $X$-axis, and the $X$-axis itself. That triangle has an acute angle —the reference angle— whose ratios match, apart from the sign, those of the original angle.
The figure shows the angle $135º$ on the unit circle, together with its reference triangle. Given that $\sin 45º = \frac{\sqrt{2}}{2}$ and…
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…
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).
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Express the bearing of the sailboat as a mathematical angle $\alpha$ between $0º$ and $360º$.
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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 $\alpha$…
Consider the angle $\alpha = 215º$.
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State which quadrant its terminal side lies in.
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Determine the sign of $\sin \alpha$, $\cos \alpha$ and $\tan \alpha$.
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Write $\alpha$ as $180º$ plus an acute angle, and also as a negative angle between $-360º$ and $0º$.
What If the Tangent Does Not Exist?
The tangent is defined as $\tan\ \alpha = \dfrac{\sin\ \alpha}{\cos\ \alpha}$. Like any division, it fails when the denominator is $0$: when $\cos\ \alpha = 0$, the tangent does not exist. This happens exactly at $90º$ and $270º$, the two points where the terminal side lies on the $Y$-axis.
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…
Determine which quadrant each of the angles $\alpha$, $\beta$ and $\gamma$ lies in, knowing that:
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$\sin \alpha < 0$ and $\tan \alpha > 0$
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$\cos \beta > 0$ and $\sin \beta < 0$
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$\cos \gamma < 0$ 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…
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 $750º$.
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How many full turns has the capsule made?
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What angle less than $360º$ describes its final position relative to its starting position?
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If the rotation had been $1500º$, where would the capsule end up?
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…
State whether each of the following statements is true or false, giving reasons for your answer:
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There exists an angle $\alpha$ such that $\cos \alpha = 1.4$.
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If $\sin \alpha > 0$, then $\alpha$ belongs to the first quadrant.
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The tangent of $90º$ does not exist.
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$\sin(-40º) = \sin 40º$.