What is trigonometry about?
“tri” - “gon” -o- “metry” is Greek for the measurement of three angles, the angles in any triangle.
So, in school, trigonometry introduces some more tools to calculate the sides, angles and areas of triangles by extending what we know about the ratios in similar right-angled triangles and linking them to the acute angles.
sine - der Sinus 1
the ratio of the side opposite to the angle and the hypotenuse is called sine
$${\sin{\alpha}= \frac{y}{h}}$$
where y is the opposite side and h is the hypotenuse.
NOTE - brackets in German text books and often none in English text books
The sine is a function of the angle in a right-angled triangle. The ratio of the opposite side and the hypotenuse has a specific angle assigned to it.
What can we do with this? Table or calculator
Knowing the angle $\alpha$ we can find the ratio from tables or with a calculator, using the $\color{orange}{[\sin]}$ button.
example
$\alpha = 17°$and$\sin{(17°)} \approx 0.29237170$to 8 decimal places
And, if we know the ratio we can read the tables in reverse and find the angle that matches the ratio. In the calculator, use the $\color{orange}{[\sin^{-1}]}$ button. It is the inverse function of sine, also known as the arcus sinus - $\color{orange}{[\arcsin]}$.
cosine - der Kosinus
the ratio of the side adjacent to the angle and the hypotenuse is called cosine
$${\cos{\alpha}= \frac{x}{h}}$$
where x is the adjacent side and h is the hypotenuse.
Use the $\color{orange}{[\cos]}$ button in a calculator. And, $\color{orange}{[\cos^{-1}]}$ for finding the angle given the ratio $\frac{\text{adjacent}}{\text{hypotenuse}}$.
tangent - der Tangens
the ratio of the opposite and the adjacent is called the tangent
$${\tan{\alpha}= \frac{\sin{\alpha}}{\cos{\alpha}}= \frac{y}{x} }$$
example
On the side of the road the sign showing the slope reads 18%. What does this mean? What is the angle of inclination?
$${18 \text{%} = \frac{ \text{18 m altitude}}{ \text{100 m horizontal distance}} = 0.18 = \tan{\alpha}}$$
from the table we know that $\alpha$ lies between 10 and 11 degrees. More acurately, using the unit circle app, it is about $\alpha = 10.2°$
Or, using a more precise table where the angles are accurate to 1 decimal place.
Use the $\color{orange}{[\tan^{-1}]}$ button on the calculator, to get approximately 10.203973722° (careful! - make sure the calculator is in degree mode - mode DEG and not RAD or GON)
finding missing measurements in a right-angled triangle
With Pythagoras’ theorem you are able to find the missing side if you know the 2 other sides of a right-angled triangle.
While, with the help of the trigonometry it is now possible to find
- the missing sides with hypotenuse and angle given
- the hypotenuse with opposite side and angle given
- the hypotenuse with adjacent side and angle given
- the acute angles of a right-angled triangle given two sides
example calculations on a right-angled triangle
find the missing sides given an angle and one side
CIMT - using the trigonometric ratios
calculating the angle
more exercises and info on Aufgabenfuchs
measurements in any triangle
Pythagoras revisited
we know from Pythagoras’ theorem that $${x^2 + y^2 = h^2}$$
from the ratios we also know that ${x = h \cdot \cos{\alpha}}$ and ${y = h \cdot \sin{\alpha}}$
So, $${x^2 = (h \cdot \cos{\alpha})^2}$$ $${y^2 = (h \cdot \sin{\alpha})^2}$$
Adding the 2 equations:
$${x^2 + y^2 = (h \cdot \cos{\alpha})^2 + (h \cdot \sin{\alpha})^2}$$
$${x^2 + y^2 = h^2 \cdot \cos^2{\alpha} + h^2 \cdot \sin^2{\alpha}}$$
$${x^2 + y^2 = h^2 \cdot \left(\cos^2{\alpha} + \sin^2{\alpha}\right)}$$
But, we know that $${x^2 + y^2 = h^2}$$ (Pythagoras’ theorem) so it follows that
$${\sin^2{\alpha} + \cos^2{\alpha} = 1}$$
cotangent - der Kotangens
the reciprocal of the tangent is the cotangent
$${\cot{\alpha}= \frac{1}{\tan{\alpha}}= \frac{\cos{\alpha}}{\sin{\alpha}}= \frac{x}{y} }$$
history
ancient Greece
India
Here is a reference to Indian mathematics while Europe was in the dark ages
not so commonly used
secant - der Sekans
the reciprocal of the cosine is the secant - read “sec”
$${\sec{\alpha}= \frac{1}{\cos{\alpha}}}$$
cosecant - der Kosekans
the reciprocal of the sine is the cosecant - read “cosec”
$${\csc{\alpha}= \frac{1}{\sin{\alpha}}}$$
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Meaning of sine: trigonometric function, 1590s (in Thomas Fale’s “Horologiographia, the Art of Dialling”), from Latin sinus “fold in a garment, bend, curve, bosom” (see sinus). Used mid-12c. by Gherardo of Cremona in Medieval Latin translation of Arabic geometrical text to render Arabic jiba “chord of an arc, sine” (from Sanskrit jya “bowstring”), which he confused with jaib “bundle, bosom, fold in a garment.” from https://www.etymonline.com/word/Sine ↩︎