Geometry and Trigonometry · Trigonometry of general angles
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-angled triangle. But a GPS needs to work out the angle of a bend of $200º$, a Ferris wheel turns 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, however large, small or negative it may be. That is the unit circle.
An angle does not have to stay between $0º$ and $360º$. If something keeps turning, 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 anticlockwise and negative if it is clockwise.
Practice★☆☆Not started
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 $360º$:
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:
Draw a circle of radius $1$ centred 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 $P$. This 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-angled triangle one: the radius of the circle (the hypotenuse) is $1$, so opposite side divided by hypotenuse is simply the $y$-coordinate of $P$.
Practice★☆☆Not started
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:
The axes divide the unit circle into four quadrants, numbered anticlockwise from the positive $X$-axis. The quadrant in which the terminal side of $\alpha$ lies determines, without any calculation, whether $\sin\ \alpha$ and $\cos\ \alpha$ are positive or negative: you just need to look at the sign of the $y$- and $x$-coordinates in that region.
Practice★☆☆Not started
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:
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:
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 $P$.
How do we find, for example, $\sin\ 135º$ without a calculator? The key is the reference triangle: the right-angled triangle formed by the terminal side, its projection onto the $X$-axis and the $X$-axis itself. This triangle has an acute angle, the reference angle, whose ratios are the same as those of the original angle, apart from the sign.
Practice★★☆Not started
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).
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 $\alpha$…
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 $0º$.
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.
Practice★★☆Not started
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:
$\sin \alpha < 0$ 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…
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?
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:
There exists an angle $\alpha$ such that $\cos \alpha = 1,4$.
If $\sin \alpha > 0$, then $\alpha$ belongs to the first quadrant.
Back to the start: The $750º$ rotation of the Ferris wheel is equivalent to $2$ complete turns plus $30º$ ($750º = 2 \cdot 360º + 30º$), so the height of the cabin relative to the centre is proportional to $\sin\ 750º = \sin\ 30º = 0{,}5$ times the radius.
Summary
Let's practice! 8 medium exercises in this section