For most of us, Pythagorean Theorem is the gateway to trigonometry.
We start with the quest for hypotenuse.

Pythagoras Hypotenuse Diagram

The length of hypotenuse is:

Pythagoras Hypotenuse Expression

Typing the formula is tedious. It’s easier to use the hypotenuse function. Here is the button.

 h 

Its shortcut is H.

Enter  3   h   4

Pythagoras Hypotenuse Display

Later you will learn  h  has another purpose.
For now, let’s meet the ‘sister’ of hypotenuse.


Adjacent

Sometimes we know the hypotenuse but we need to find the other side, for example:

Pythagoras Adjacent Diagram

The length of adjacent is:

Pythagoras Adjacent Expression

The adjacent function makes this easy.
It is the secondary function of  h , accessible via shift (⇧).

=  h  =  ⇧H

Enter  5 3

Pythagoras Adjacent Display

Adjacent is not fussy about the order, you can also enter  3 5.

Let’s calculate the height of a triangle.

Basics Sine Diagram

From the definition of sine, the height is

8 × sin 30°

Or more concisely 
8 sin 30°

So you enter:

8 sin 30

Basics Sine Display

By default, angles are in degrees, hence 30 is displayed as 30°.

The Angle button is both a switch and an indicator.

DEG

It switches between degrees and radians. When it indicates DEG,
your angles are in degrees; otherwise RAD for radians.


The shortcuts are quite easy to remember.

sin S
cos C
tan T

For 8 sin 30, you can type:

8  S  30


Inverse

Sometimes we need to find the angle. For example:

Basics Inverse Sine Diagram

We know that

sin ? = 4/8 = 0.5

Therefore the angle is the inverse sine or the arcsin of 0.5.
We write it as  sin–1 0.5.


Entering sin–1 — The traditional way

Mouse: Click follow by sin
Keyboard: Hold shift and press S  at the same time

This method is common amongst traditional calculators.
However it’s a bit awkward to press two keys at once.


Entering sin–1 — The natural way

Let’s look at it visually.

Inverse Sin Entity

sin–1 has 2 parts.

Inverse Sin Components

It turns out there are buttons for them.

sin sin
inverse x–1

All you need is to enter sin and –1.

Mouse:

Inverse Sin Via Mouse

Keyboard:

Inverse Sin Via Keyboard

The shortcut for inverse is the single quote ( ' ).
Yes you can think visually too.

How to enter

We write hyperbolic sine as:

Sinh Entity

Again think of it as 2 parts:

Sinh Components

Therefore you enter sin, then h.

Mouse:

Sinh Via Mouse

Keyboard:

Sinh Via Keyboard

The inverse is similar.

Sinh Inv Entity

These are its parts:

Sinh Inv Components

This is how you enter:

Mouse:

Sinh Inv Via Mouse

Keyboard:

Sinh Inv Via Keyboard

These are cosecant (csc), secant (sec), cotangent (cot).
There are 2 ways to enter.


The easy way — via mouse

Click and hold the button sin, cos, or tan:

Sine Button Sine Menu

The efficient way — via keyboard

The is quite interesting. More rewarding too.

First, let’s meet our new friend, inverse transformation.
It’s under the menu Calculation ▸ Inverse.  Shortcut: ⌘ '

You can use it to invert a number.
E.g.  enter 4 and press ⌘ ' for 1/4. You get 0.25.

4 inverts to 0.25

Now let’s get back to csc, sec, cot.
Knowing that they are reciprocals of sin, cos, and tan…

1/sin = csc, 1/cos = sec, 1/tan = cot

And inverse transformation can give us the reciprocal…

4 inverted to 1/4

Yes you can invert trigonometric functions:

sine inverted to cosecant

So to enter csc, press:

sin  ⌘ '

Since the shortcut for sin is S, you can simply enter:

S  ⌘ '

Working in degrees is easier because we are dealing with simple numbers. For example:

Arcsin 1 Degrees

In contrast with radians, we face with messy numbers.

Arcsin 1 Radians 1.57

In degrees, 360° is one revolution, so it’s easy to see 90° is ¼ revolution.
In radians, it’s hard to see 1.570796 is one half of π.


But if we view the result as fraction, Magic Number will show radians as a fraction of π by default. ( Shortcut: ⌘ / )

Fraction Popover
Arcsin 1 Radians 1 2 Pi

If you like, you can see decimal radians with π.
Click and hold the RAD button to see its menu, then choose ‘r × π’.

Angle Button Angle Menu
Arcsin 1 Radians 0.5 Pi

Introducing tau, τ

While ½ π is more approachable than 1.570796, still we have to do the extra arithmetic to see this angle naturally as ¼ revolution.

This is where tau can help because
τ  =  2 π

That is,
1 τ  radians  =  1 revolution

This makes radians more intuitive, more so than degrees:

Tau diagram

We can go on. But we will let these smart people do the talking.


To use τ as the default, click and hold the π button

π Button π Menu

From now on, radians will be expressed in terms of τ.

Arcsin 1 Radians 1/4 Tau

And in decimal form:

Arcsin 1 Radians 0.25 Tau

Entering angles in radians

45° is 0.785398 radians. In practice it’s quicker and more accurate to type it as π / 4 or τ / 8.

ConstantNameShortcut
πPiP
τTau⌥T

Generally one has to put parentheses around them, for instance, “tan (π / 4)”.

Tan Pi Over 4 Bracketed

As π and τ are angular constants, if you use them for trigonometry, parentheses are not required.

The simple way:

Tan Pi Over 4
Tan Tau Over 8

Working with degrees and radians

You can switch between degrees and radians via the Angle button.

DEG  ⟷  RAD

Shortcut: ⌥⌘R.

If you want to convert one unit to another, go to menu
Calculation ▸ Extra Functions ▸ Trigonometry (⇧⌘F).


Entering degrees, minutes, seconds

Use the time format ‘hour : minute : second’.
It is analogous to degree° minute′ second″

For sin 30°40′50″, enter sin 30:40:50.

sin 30:40:50° = 0.5102510808

The ‘Double Key’ technique

Buttons like sin has a secondary function, sin–1, you access it by pressing its shortcut while holding shift (⇧). It’s a two-finger hassle.

+S sin–1

With Double Key, you can do it with one finger — just hit S twice.

SS sin–1

Likewise for cos–1 and tan–1.

CC cos–1

TT tan–1

And for the adjacent function you learned earlier:

HH adjacent


Embrace the unknown

We spent a lot of time talking about the inverse. The truth is, you don’t have to use it.

Earlier we calculated an angle with sin–1 0.5, you get achieve the same by entering sin ? = 0.5.

Sin Of Unknown

? is the unknown, just like ‘x’ in algebra.

Here is an example with the law of sines:

Sine Rule Diagram
Sine Rule Equation

Click here to learn more about the unknown.