Intro to TRIGONOMETRY

$\sin^2\theta + \cos^2\theta = 1$

38.2 Pythagoras Theorem

The Pythagorean Theorem describes the relationship between the three sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).

$$ a^2 + b^2 = c^2 $$

Where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

Pythagoras Solver

Side $a$ (optional):

Side $b$ (optional):

Side $c$ (hypotenuse, optional):

Describe your Pythagoras problem, e.g., 'A right triangle has legs of 3 and 4, find the hypotenuse.'

38.3 Sine, Cosine, Tangent (SOH CAH TOA)

These are the fundamental trigonometric ratios for right-angled triangles. They relate the angles of a triangle to the lengths of its sides.

Sine (SOH): $ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $
Cosine (CAH): $ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $
Tangent (TOA): $ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $

Where Opposite, Adjacent, and Hypotenuse refer to the sides relative to the angle $\theta$ in a right-angled triangle.

SOH CAH TOA Solver

Angle $\theta$ (degrees):

Opposite Side (optional):

Adjacent Side (optional):

Hypotenuse (optional):

Describe your SOH CAH TOA problem, e.g., 'In a right triangle, angle A is 30 degrees and the hypotenuse is 10. Find the opposite side.'

38.4 Evaluating Trig Ratios (Acute Angles)

Evaluating trigonometric ratios means finding the numerical value of $\sin(\theta)$, $\cos(\theta)$, or $\tan(\theta)$ for a given acute angle $\theta$. These values can be found using a calculator or, for special angles, from exact value tables.

Trig Ratio Evaluator

Angle $\theta$ (degrees):

Describe your problem, e.g., 'Evaluate sin(45 degrees) and explain.'

38.5 Reciprocal Ratios

In addition to sine, cosine, and tangent, there are three reciprocal trigonometric ratios:

Cosecant (csc): $ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}} $
Secant (sec): $ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}} $
Cotangent (cot): $ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}} $

Reciprocal Ratio Solver

Angle $\theta$ (degrees):

Describe your problem, e.g., 'Find cosec(60 degrees).''

38.6 Fractional & Surd Forms (Exact Values)

For certain "special" angles ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$), the trigonometric ratios have exact values that can be expressed as fractions or surds (square roots), without needing a calculator.

Exact Values Table

Ratio	0° ($0$ rad)	30° ($\pi/6$ rad)	45° ($\pi/4$ rad)	60° ($\pi/3$ rad)	90° ($\pi/2$ rad)
$\sin(\theta)$	$0$	$\\frac{1}{2}$	$\\frac{\\sqrt{2}}{2}$	$\\frac{\\sqrt{3}}{2}$	$1$
$\cos(\theta)$	$1$	$\\frac{\\sqrt{3}}{2}$	$\\frac{\\sqrt{2}}{2}$	$\\frac{1}{2}$	$0$
$\tan(\theta)$	$0$	$\\frac{1}{\\sqrt{3}}$ (or $\\frac{\\sqrt{3}}{3}$)	$1$	$\\sqrt{3}$	Undefined

Describe your problem, e.g., 'What is the exact value of tan(60 degrees)?'

38.7 Solving Right-Angled Triangles

Solving a right-angled triangle means finding the measures of all its sides and angles when some information is given. This typically involves using the Pythagorean Theorem and/or the SOH CAH TOA ratios.

Right-Angled Triangle Solver

Enter at least two values (one must be a side) to solve the triangle.

Angle A (degrees, not 90):

Angle B (degrees, not 90):

Side $a$ (opposite Angle A):

Side $b$ (opposite Angle B):

Side $c$ (hypotenuse):

Describe your triangle problem, e.g., 'A right triangle has an angle of 40 degrees and the adjacent side is 7. Solve the triangle.'

38.8 Elevation & Depression Angles

These angles are used in real-world applications of trigonometry, often involving heights and distances.

Angle of Elevation: The angle measured upwards from the horizontal line of sight to an object above.
Angle of Depression: The angle measured downwards from the horizontal line of sight to an object below.

Elevation/Depression Solver

Calculate height or distance given angle and one side.

Angle (degrees):

Horizontal Distance (Adjacent side):

Vertical Height (Opposite side):

Describe your problem, e.g., 'From 100m away, the angle of elevation to the top of a building is 35 degrees. What is the height of the building?'

38.9 Small Angle Approximations

For very small angles $\theta$ (in radians), we can use the following approximations:

$ \sin(\theta) \approx \theta $
$ \cos(\theta) \approx 1 - \frac{\theta^2}{2} $
$ \tan(\theta) \approx \theta $

Important: These approximations are only valid when $\theta$ is measured in radians.

Small Angle Approximator

Small Angle $\theta$ (radians):

Describe your problem, e.g., 'Explain small angle approximation for sin(theta).'