Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles.
• It's fundamental to fields like physics, engineering, astronomy, and surveying.
• Trigonometric functions are used to relate the angles of a triangle to the ratios of its sides.
• Sine, cosine, and tangent are the primary trigonometric functions.
• Cosecant, secant, and cotangent are their reciprocals.

Right-Angled Triangles

• Trigonometry primarily deals with right-angled triangles.
• Right-angled triangles have one angle that measures 90 degrees.
• The side opposite the right angle is the hypotenuse, which is the longest side.
• The other two sides are referred to as the opposite and adjacent sides, relative to a specific angle.

Trigonometric Ratios

• Sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse: sin(θ) = Opposite / Hypotenuse.
• Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: cos(θ) = Adjacent / Hypotenuse.
• Tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side: tan(θ) = Opposite / Adjacent.
• Cosecant (csc) is the reciprocal of sine: csc(θ) = Hypotenuse / Opposite.
• Secant (sec) is the reciprocal of cosine: sec(θ) = Hypotenuse / Adjacent.
• Cotangent (cot) is the reciprocal of tangent: cot(θ) = Adjacent / Opposite.
• The mnemonic SOH CAH TOA is often used to remember these ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Common Angles

• Certain angles occur frequently in trigonometry.
• It's useful to know the trigonometric function values for 0°, 30°, 45°, 60°, and 90°.
• sin(0°) = 0, cos(0°) = 1, tan(0°) = 0
• sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
• sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1
• sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3
• sin(90°) = 1, cos(90°) = 0, tan(90°) = undefined

Pythagorean Theorem

• The Pythagorean Theorem is crucial in trigonometry.
• It states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a² + b² = c².
• This theorem is used to find the length of an unknown side of a right-angled triangle if the lengths of the other two sides are known.

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.
• The Pythagorean identity: sin²(θ) + cos²(θ) = 1.
• Variations of the Pythagorean identity: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ).
• Quotient identities: tan(θ) = sin(θ) / cos(θ) and cot(θ) = cos(θ) / sin(θ)
• Reciprocal Identities: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ)
• These identities are used to simplify trigonometric expressions and solve equations.

Angle of Elevation and Depression

• The angle of elevation is the angle between the horizontal line and the line of sight when an observer looks upward to a point.
• The angle of depression is the angle between the horizontal line and the line of sight when an observer looks downward to a point.
• These angles are used in problems involving heights and distances.

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
• It is used to define trigonometric functions for all real numbers (angles).
• For any angle θ, the coordinates of the point where the terminal side of the angle intersects the unit circle are (cos(θ), sin(θ)).
• This allows for the definition of trigonometric functions for angles greater than 90° and negative angles.

Sine and Cosine Rules

• Sine Rule: a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the side lengths of a triangle and A, B, and C are the angles opposite those sides, respectively.
• Cosine Rule: c² = a² + b² - 2ab cos(C), where a, b, and c are the side lengths of a triangle and C is the angle opposite side c.
• These rules are used to solve triangles when not right-angled.

Applications

• Used in navigation to determine direction and position.
• Used in surveying to measure distances and heights.
• Used in physics to analyze wave motion, optics, and mechanics.
• Used in engineering to design structures, analyze forces, and model systems.
• Used in astronomy to calculate distances to stars and planets.

Solving Trigonometric Equations

• Trigonometric equations involve trigonometric functions.
• To solve them, use algebraic techniques and trigonometric identities to isolate the trigonometric function.
• Consider the period of the trigonometric function and find all solutions within that period.
• General solutions can be expressed by adding integer multiples of the period to the solutions found within one period.

Inverse Trigonometric Functions

• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle that corresponds to a given trigonometric ratio.
• arcsin(x) or sin⁻¹(x) gives the angle whose sine is x.
• arccos(x) or cos⁻¹(x) gives the angle whose cosine is x.
• arctan(x) or tan⁻¹(x) gives the angle whose tangent is x.
• The domains and ranges of inverse trigonometric functions are restricted to ensure they are single-valued.