Trigonometric functions

Sine, cosine and tangent beyond 90° Extended

In a right-angled triangle, $\sin$, $\cos$ and $\tan$ only make sense for acute angles. To give them a value for any angle, mathematicians use a circle of radius 1 centred on the origin.

Start at the point $(1,0)$ and turn anticlockwise about the origin through an angle of $x^{°}$. The point you reach has coordinates

$$(\cos x,\sin x)$$

For an acute angle this agrees with the triangle definitions: drop a perpendicular to the $x$-axis and you get a right-angled triangle with hypotenuse 1, so the adjacent side is $\cos x$ and the opposite side is $\sin x$. The tangent is then $\tan x=\frac{\sin x}{\cos x}$.

As the point goes round the circle, its coordinates change sign:

• from $0^{°}$ to ${90}^{°}$ the point is above and to the right of the origin: $\sin$, $\cos$ and $\tan$ are all positive;
• from ${90}^{°}$ to ${180}^{°}$ it is above and to the left: $\sin x$ is positive, $\cos x$ and $\tan x$ are negative;
• from ${180}^{°}$ to ${270}^{°}$ it is below and to the left: $\sin x$ and $\cos x$ are negative, but $\tan x$ is positive (a negative divided by a negative);
• from ${270}^{°}$ to ${360}^{°}$ it is below and to the right: $\cos x$ is positive, $\sin x$ and $\tan x$ are negative.

After ${360}^{°}$ the point is back where it started, so the values repeat. This is why the graphs below are waves.

The graph of y = sin x Extended

The sine of the angle is the height of the point on the circle. It starts at 0, rises to 1 when the point is at the top (${90}^{°}$), falls back to 0 at ${180}^{°}$, reaches $-1$ at the bottom (${270}^{°}$) and returns to 0 at ${360}^{°}$.

Features to know:

• it passes through $(0,0)$, $(180,0)$ and $(360,0)$;
• maximum value 1 at $x=90$; minimum value $-1$ at $x=270$;
• every value of $\sin x$ lies between $-1$ and 1;
• it repeats every ${360}^{°}$.

Symmetry. The arch from $0^{°}$ to ${180}^{°}$ is symmetrical about the line $x=90$. So an angle and its supplement have the same sine:

$$\sin ({180}^{°}-x)=\sin x$$

For example, $\sin {150}^{°}=\sin {30}^{°}=\frac{1}{2}$. The part from ${180}^{°}$ to ${360}^{°}$ is the same arch upside down, so $\sin ({180}^{°}+x)=-\sin x$ and $\sin ({360}^{°}-x)=-\sin x$.

The graph of y = cos x Extended

The cosine is the horizontal coordinate of the point on the circle. It starts at 1, falls to 0 at ${90}^{°}$, reaches $-1$ at ${180}^{°}$, is back to 0 at ${270}^{°}$ and returns to 1 at ${360}^{°}$. It is the sine curve shifted ${90}^{°}$ to the left.

Features to know:

• it passes through $(0,1)$, $(90,0)$, $(180,-1)$, $(270,0)$ and $(360,1)$;
• every value of $\cos x$ lies between $-1$ and 1;
• it repeats every ${360}^{°}$.

Symmetry. The curve is symmetrical about the line $x=180$, so

$$\cos ({360}^{°}-x)=\cos x$$

For example, $\cos {300}^{°}=\cos {60}^{°}=\frac{1}{2}$. Also, $\cos ({180}^{°}-x)=-\cos x$: the cosine of an obtuse angle is negative, which you will use with the cosine rule.

The graph of y = tan x Extended

Since $\tan x=\frac{\sin x}{\cos x}$, the tangent is 0 wherever $\sin x=0$, and it cannot be worked out wherever $\cos x=0$, at ${90}^{°}$ and ${270}^{°}$. As $x$ gets close to ${90}^{°}$ from below, $\cos x$ is a tiny positive number, so $\tan x$ becomes enormous. Just past ${90}^{°}$, $\cos x$ is a tiny negative number and $\tan x$ is hugely negative.

Features to know:

• it passes through $(0,0)$, $(180,0)$ and $(360,0)$, and through $(45,1)$ and $(225,1)$;
• it has asymptotes, vertical lines the curve approaches but never meets, at $x=90$ and $x=270$;
• it takes every value, however large, positive or negative;
• it repeats every ${180}^{°}$, not every ${360}^{°}$.

Symmetry. Because the pattern repeats every ${180}^{°}$,

$$\tan ({180}^{°}+x)=\tan x$$

For example, $\tan {225}^{°}=\tan {45}^{°}=1$.

Sketching the graphs Extended

A sketch is not a plotted graph, but it must show the right shape in the right place. An examiner looks for:

• axes with $x$ marked at $90$, $180$, $270$ and $360$, and $y$ marked at 1 and $-1$ (for $\tan$, just the shape and the asymptotes);
• a smooth curve through the key points: a rounded wave, not a zigzag of straight lines, and not pointed at the top;
• the correct start: $\sin$ starts at the origin going up, $\cos$ starts at $(0,1)$ going down, $\tan$ starts at the origin going up;
• for $\tan x$, the asymptotes at $x=90$ and $x=270$ drawn as dashed lines, with the curve getting steeper as it approaches them but never touching or crossing them.

Keep the sketch inside $0⩽x⩽360$ unless you are asked for more.

Interpreting the graphs Extended

The solutions of $\sin x=k$ are the $x$-coordinates of the points where the horizontal line $y=k$ meets the curve $y=\sin x$. So the graph tells you how many solutions there are before you calculate any of them.

• If $-1<k<1$, the line $y=k$ crosses the sine curve twice in $0⩽x⩽360$ (except $k=0$, which meets it at 0, 180 and 360), and the same for cosine.
• If $k=1$ or $k=-1$ the line just touches the curve at a maximum or minimum: one solution (for $\cos x=1$ there are two, at 0 and 360).
• If $k>1$ or $k<-1$ the line misses the curve completely: there are no solutions. $\sin x=1.5$ is impossible.
• The line $y=k$ crosses each branch of $y=\tan x$ once, so $\tan x=k$ always has two solutions in this range (three when $k=0$).

The graph also shows where the second solution is, by symmetry. That is the idea behind every equation in the next section.

Solving trigonometric equations Extended

Your calculator's ${\sin }^{-1}$, ${\cos }^{-1}$ and ${\tan }^{-1}$ keys give one angle only, called the principal value:

• ${\sin }^{-1}$ gives an angle from $-{90}^{°}$ to ${90}^{°}$;
• ${\cos }^{-1}$ gives an angle from $0^{°}$ to ${180}^{°}$;
• ${\tan }^{-1}$ gives an angle from $-{90}^{°}$ to ${90}^{°}$.

The equation usually has another solution in $0⩽x⩽360$, and the calculator will never show it. You find it from the symmetry of the graph.

Method.

1. Rearrange the equation so that $\sin x$, $\cos x$ or $\tan x$ is on its own: $\sin x=k$.
2. For $\sin$ or $\cos$, check that $k$ is between $-1$ and 1. If it is not, there are no solutions.
3. Find the principal value $p$: with the inverse key, or from the exact values you know.
4. Find the second solution:

• $\sin x=k$: $x=p$ and $x={180}^{°}-p$;
• $\cos x=k$: $x=p$ and $x={360}^{°}-p$;
• $\tan x=k$: $x=p$ and $x=p+{180}^{°}$.

1. If a value is outside $0⩽x⩽360$, add or subtract ${360}^{°}$ (or ${180}^{°}$ for $\tan$) to bring it into the range.
2. Give each answer to 1 decimal place unless it is exact, and check one by substituting it back.

When $k$ is negative, it often helps to work with the acute angle $\alpha$ for which $\sin \alpha$ (or $\cos \alpha$, $\tan \alpha$) equals $\left|k\right|$, and then place it using the signs: sine is negative between 180 and 360, so $x=180+\alpha$ or $360-\alpha$; cosine is negative between 90 and 270, so $x=180-\alpha$ or $180+\alpha$; tangent is negative between 90 and 180 and between 270 and 360, so $x=180-\alpha$ or $360-\alpha$.

On the non-calculator paper the value of $k$ will be one of the exact values: $\sin {30}^{°}=\frac{1}{2}$, $\sin {45}^{°}=\frac{\sqrt{2}}{2}$, $\sin {60}^{°}=\frac{\sqrt{3}}{2}$, $\cos {60}^{°}=\frac{1}{2}$, $\cos {30}^{°}=\frac{\sqrt{3}}{2}$, $\tan {45}^{°}=1$, $\tan {60}^{°}=\sqrt{3}$, $\tan {30}^{°}=\frac{1}{\sqrt{3}}$, together with the values at $0^{°}$, ${90}^{°}$, ${180}^{°}$, ${270}^{°}$ and ${360}^{°}$ read from the graphs.