Trigonometry | Elective Mathematics | The Let's Learn Project

Introduction to Trigonometry

Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles. The word comes from Greek "trigonon" (triangle) and "metron" (measure).

Right-Angled Triangle:

Hypotenuse (h)

Opposite (o)

Adjacent (a)

Applications:

Surveying and navigation
Engineering and construction
Physics and astronomy
Computer graphics

Trigonometric Ratios

The three primary trigonometric ratios relate the sides of a right-angled triangle to an acute angle:

Sine (sin)

\[ \sin θ = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{o}{h} \]

Cosine (cos)

\[ \cos θ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{h} \]

Tangent (tan)

\[ \tan θ = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{o}{a} \]

Example:

For a right-angled triangle with opposite = 3, adjacent = 4, hypotenuse = 5:

\[ \sin θ = \frac{3}{5}, \quad \cos θ = \frac{4}{5}, \quad \tan θ = \frac{3}{4} \]

                                    
                                         Mnemonic: SOH-CAH-TOA
                                    
                                    SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

Reciprocal Ratios

Cosecant (csc)

\[ \csc θ = \frac{1}{\sin θ} = \frac{h}{o} \]

Secant (sec)

\[ \sec θ = \frac{1}{\cos θ} = \frac{h}{a} \]

Cotangent (cot)

\[ \cot θ = \frac{1}{\tan θ} = \frac{a}{o} \]

Trigonometric Functions

Trigonometric functions extend the concept of ratios to all angles using the unit circle (radius = 1):

(x, y)

For any angle θ:

\[ \sin θ = y \] \[ \cos θ = x \] \[ \tan θ = \frac{y}{x} \]

Periodicity:

Trigonometric functions repeat every 360° (2π radians):

\[ \sin(θ + 360°) = \sin θ \] \[ \cos(θ + 360°) = \cos θ \] \[ \tan(θ + 180°) = \tan θ \]

Graphs:

Sine

Cosine

Tangent

Pythagorean Identities

\[ \sin^2θ + \cos^2θ = 1 \]

\[ 1 + \tan^2θ = \sec^2θ \]

\[ 1 + \cot^2θ = \csc^2θ \]

Angles of Elevation and Depression

Angle of Elevation

Angle between horizontal and line of sight when looking upward

Angle of Depression

Angle between horizontal and line of sight when looking downward

Example Problem:

A 20m tall building casts a shadow 15m long. Find the angle of elevation of the sun.

Using tangent ratio:

\[ \tan θ = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{20}{15} \] \[ θ = \tan^{-1}\left(\frac{4}{3}\right) ≈ 53.13° \]

Special Angles

Certain angles have exact trigonometric values that can be derived geometrically:

Angle (θ)	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	∞

Derivation of 45° Values:

Consider an isosceles right triangle with legs = 1:

\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \] \[ \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] \[ \tan 45° = \frac{1}{1} = 1 \]

Quadrantal Angles

Angles that lie on the x-axis or y-axis (0°, 90°, 180°, 270°, 360°):

I II III IV

0°

90°

180°

270°

Signs in Quadrants:

I: All positive
II: sin positive
III: tan positive
IV: cos positive

Mnemonic: All Students Take Calculus

Reference Angles:

For any angle θ, its reference angle is the acute angle between the terminal side and the x-axis:

Quadrant	Reference Angle
I	θ
II	180° - θ
III	θ - 180°
IV	360° - θ

Radian Measurement

An alternative to degree measurement where angles are defined by arc length:

\[ 1 \text{ radian} = \frac{180°}{π} ≈ 57.2958° \]

1 rad

1 radian is the angle subtended by an arc equal in length to the radius

Conversions:

\[ π \text{ radians} = 180° \] \[ 1° = \frac{π}{180} \text{ radians} \] \[ 1 \text{ radian} = \frac{180°}{π} \]

Common Conversions:

Degrees	Radians
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
360°	2π

Example Conversions:

Convert 150° to radians:

\[ 150° × \frac{π}{180°} = \frac{5π}{6} \text{ radians} \]

Convert 3π/4 radians to degrees:

\[ \frac{3π}{4} × \frac{180°}{π} = 135° \]

Practice Exercise

Question 1

Find the exact value of sin 60° cos 30° + cos 60° sin 30°

Using special angles:

\[ \frac{\sqrt{3}}{2} × \frac{\sqrt{3}}{2} + \frac{1}{2} × \frac{1}{2} = \frac{3}{4} + \frac{1}{4} = 1 \]

Question 2

A ladder 5m long leans against a wall, making a 60° angle with the ground. How far up the wall does it reach?

\[ \sin 60° = \frac{\text{Height}}{5} \] \[ \text{Height} = 5 × \frac{\sqrt{3}}{2} ≈ 4.33 \text{ m} \]

Question 3

Convert 5π/12 radians to degrees and find its reference angle.

\[ \frac{5π}{12} × \frac{180°}{π} = 75° \]

75° is in Quadrant I, so reference angle = 75°

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