Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles

Trigonometric Functions

• Sine (sin), cosine (cos), and tangent (tan) are primary trigonometric functions
• These functions relate angles of a right triangle to the ratios of two of its sides
• Cosecant (csc), secant (sec), and cotangent (cot) are reciprocal trigonometric functions
• Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent
• Trigonometric functions are used extensively in various fields such as physics, engineering, and navigation

Right Triangles

• A right triangle includes one angle of 90 degrees
• The side opposite the right angle is the hypotenuse, the longest side of the triangle
• The other two sides are called legs (or cathetus)
• The leg opposite to the angle under consideration is called "opposite"
• The leg adjacent to the angle under consideration is called "adjacent"

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) of an angle is the ratio of the length of the hypotenuse to the length of the opposite side: csc(θ) = Hypotenuse / Opposite = 1 / sin(θ)
• Secant (sec) of an angle is the ratio of the length of the hypotenuse to the length of the adjacent side: sec(θ) = Hypotenuse / Adjacent = 1 / cos(θ)
• Cotangent (cot) of an angle is the ratio of the length of the adjacent side to the length of the opposite side: cot(θ) = Adjacent / Opposite = 1 / tan(θ)

Pythagorean Theorem

• In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides
• This is expressed as: a² + b² = c², where c is the length of the hypotenuse, and a and b are the lengths of the legs

Common Angles

• Trigonometric functions of certain angles are commonly used and should be memorized
• 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

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined
• sin²(θ) + cos²(θ) = 1 (Pythagorean identity)
• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)
• sec(θ) = 1 / cos(θ)
• csc(θ) = 1 / sin(θ)
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

Angle Sum and Difference Formulas

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double Angle Formulas

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
• tan(2θ) = (2tan(θ)) / (1 - tan²(θ))

Half Angle Formulas

• sin(θ/2) = ±√((1 - cos(θ)) / 2)
• cos(θ/2) = ±√((1 + cos(θ)) / 2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ))) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)
• The sign (±) depends on the quadrant in which θ/2 lies

Law of Sines

• In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant
• a / sin(A) = b / sin(B) = c / sin(C), where a, b, and c are side lengths, and A, B, and C are opposite angles

Law of Cosines

• Relates the lengths of the sides of a triangle to the cosine of one of its angles
• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)

Applications

• Used in surveying to calculate distances and angles
• Used in navigation to determine the course and position of ships and aircraft
• Used in physics to analyze wave motion, optics, and mechanics
• Used in engineering to design structures, machines, and electrical circuits

Inverse Trigonometric Functions

• Used to find an angle when the value of a trigonometric ratio is known
• Arcsine (sin⁻¹ or asin): Returns the angle whose sine is a given number.
• Arccosine (cos⁻¹ or acos): Returns the angle whose cosine is a given number.
• Arctangent (tan⁻¹ or atan): Returns the angle whose tangent is a given number.