5.1.1 Trigonometry - Definitions

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. Below are the key definitions and concepts in trigonometry:

1. Basic Trigonometric Ratios:

These ratios are defined for a right-angled triangle.

• Sine (sin): $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ It is the ratio of the length of the side opposite the angle $\ \theta$ to the length of the hypotenuse.
• Cosine (cos): $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ It is the ratio of the length of the side adjacent to the angle $\ \theta$ to the length of the hypotenuse.
• Tangent (tan): $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ It is the ratio of the length of the side opposite the angle $\ \theta$ to the length of the side adjacent to the angle.

2. Reciprocal Trigonometric Ratios:

• Cosecant (csc or cosec): $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$ It is the reciprocal of sine.
• Secant (sec): $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$ It is the reciprocal of cosine.
• Cotangent (cot): $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$ It is the reciprocal of tangent.

3. Unit Circle:

The unit circle is a circle with a radius of $1$, centred at the origin of a coordinate plane. The trigonometric ratios can also be defined using the unit circle:

• Sine: The $\ y$-coordinate of the point where the terminal side of the angle intersects the unit circle.
• Cosine: The $\ x$-coordinate of that point.
• Tangent: The ratio of the sine to the cosine (or $\ y/x$).

4. Pythagorean Identity:

This identity relates the square of the sine and cosine of an angle: $\sin^2 \theta + \cos^2 \theta = 1$

5. Angle Conversions:

• Degrees and Radians:
  • $\ 360^\circ = 2\pi$ radians
  • $\ 180^\circ = \pi$ radians
  • To convert from degrees to radians: $\ \text{Radians}$= $\text{Degrees} \times \frac{\pi}{180}$
  • To convert from radians to degrees: $\ \text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

6. Key Angles:

Some angles have trigonometric values that are important to remember:

• $\ 0^\circ or \ 0$ radians: $\sin 0^\circ = 0, \quad \cos 0^\circ = 1, \quad \tan 0^\circ = 0$
• $\ 30^\circ \ or \ \frac{\pi}{6}$ radians: $\ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}$
• $\ 45^\circ \ or \ \frac{\pi}{4}$ radians: $\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \tan 45^\circ = 1$
• $\ 60^\circ \ or \ \frac{\pi}{3}$ radians: $\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \tan 60^\circ = \sqrt{3}$
• $\ 90^\circ \ or \ \frac{\pi}{2}$ radians: $\sin 90^\circ = 1, \quad \cos 90^\circ = 0, \quad \tan 90^\circ \text{ is undefined}$

7. Trigonometric Identities:

• Sum and Difference Formulas: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
• Double Angle Formulas: $\sin 2A = 2\sin A \cos A$ $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

8. Inverse Trigonometric Functions:

These functions reverse the trigonometric ratios, giving the angle when the ratio is known:

• $\ \sin^{-1} x (arcsin x)$: Gives the angle whose sine is $\ x .$
• $\ \cos^{-1} x (arccos x)$: Gives the angle whose cosine is $\ x .$
• $\ \tan^{-1} x$ $(arctan x)$: Gives the angle whose tangent is $\ x$. These are defined within specific ranges to be functions.

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