Trigonometric Ratios

• Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles
• Trigonometry is fundamental to fields like engineering, physics, and navigation

Trigonometric Ratios

• Trigonometric ratios relate the angles of a right triangle to the ratios of its sides
• Consider a right triangle with one angle labeled (\theta)
• The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan)
• Sine of (\theta) [sin((\theta))] is the ratio of the length of the opposite side to the length of the hypotenuse
• Cosine of (\theta) [cos((\theta))] is the ratio of the length of the adjacent side to the length of the hypotenuse
• Tangent of (\theta) [tan((\theta))] is the ratio of the length of the opposite side to the length of the adjacent side
• There are also reciprocal trigonometric ratios: cosecant (csc), secant (sec), and cotangent (cot)
• Cosecant of (\theta) [csc((\theta))] is the reciprocal of sin((\theta)), equal to hypotenuse/opposite
• Secant of (\theta) [sec((\theta))] is the reciprocal of cos((\theta)), equal to hypotenuse/adjacent
• Cotangent of (\theta) [cot((\theta))] is the reciprocal of tan((\theta)), equal to adjacent/opposite

Common Angles

• Certain angles appear frequently in trigonometric problems
• It's useful to know the trigonometric ratios for 0°, 30°, 45°, 60°, and 90°
• sin(0°) = 0, cos(0°) = 1, tan(0°) = 0
• sin(30°) = 1/2, cos(30°) = (\sqrt{3})/2, tan(30°) = 1/(\sqrt{3})
• sin(45°) = 1/(\sqrt{2}), cos(45°) = 1/(\sqrt{2}), tan(45°) = 1
• sin(60°) = (\sqrt{3})/2, cos(60°) = 1/2, tan(60°) = (\sqrt{3})
• sin(90°) = 1, cos(90°) = 0, tan(90°) is undefined

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables
• Pythagorean identities are fundamental
• sin²((\theta)) + cos²((\theta)) = 1
• 1 + tan²((\theta)) = sec²((\theta))
• 1 + cot²((\theta)) = csc²((\theta))
• Quotient Identities
• tan((\theta)) = sin((\theta))/cos((\theta))
• cot((\theta)) = cos((\theta))/sin((\theta))
• Reciprocal Identities
• csc((\theta)) = 1/sin((\theta))
• sec((\theta)) = 1/cos((\theta))
• cot((\theta)) = 1/tan((\theta))
• Sum and Difference Formulas
• sin(A ± B) = sin(A)cos(B) ± cos(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))
• Double-Angle Formulas
• sin(2(\theta)) = 2sin((\theta))cos((\theta))
• cos(2(\theta)) = cos²((\theta)) - sin²((\theta)) = 2cos²((\theta)) - 1 = 1 - 2sin²((\theta))
• tan(2(\theta)) = (2tan((\theta)))/(1 - tan²((\theta)))
• Half-Angle Formulas
• sin((\theta)/2) = ±√((1 - cos((\theta)))/2)
• cos((\theta)/2) = ±√((1 + cos((\theta)))/2)
• tan((\theta)/2) = ±√((1 - cos((\theta)))/(1 + cos((\theta))) = sin((\theta))/(1 + cos((\theta)))=(1 - cos((\theta)))/sin((\theta))

Solving Trigonometric Equations

• Solving trigonometric equations involves finding the angles that satisfy a given equation
• Use algebraic manipulation and trigonometric identities to isolate the trigonometric function
• Consider the period of the trigonometric function when finding all possible solutions
• Check for extraneous solutions
• General solutions can be expressed as (\theta) = α + 2πk or (\theta) = (π - α) + 2πk, where k is an integer

Law of Sines

• The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles
• In any triangle ABC, a/sin(A) = b/sin(B) = c/sin(C)
• This law is useful for solving triangles when you know two angles and one side (AAS or ASA) or two sides and an angle opposite one of them (SSA)
• The SSA case can lead to ambiguous solutions (zero, one, or two triangles)

Law of Cosines

• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles
• For any triangle ABC:
• a² = b² + c² - 2bc cos(A)
• b² = a² + c² - 2ac cos(B)
• c² = a² + b² - 2ab cos(C)
• Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS)

Radians and Degrees

• Angles can be measured in degrees or radians
• A complete circle is 360 degrees or 2π radians
• To convert from degrees to radians, multiply by π/180
• To convert from radians to degrees, multiply by 180/π

Trigonometric Functions of Real Numbers

• Trigonometric functions can be defined for real numbers, not just angles
• Consider a unit circle (radius 1) in the coordinate plane
• For any real number t, start at the point (1, 0) and travel a distance of |t| along the circumference of the circle
• If t > 0, move counterclockwise; if t < 0, move clockwise
• The terminal point (x, y) on the circle determines the values of the trigonometric functions
• cos(t) = x, sin(t) = y, tan(t) = y/x

Graphs of Trigonometric Functions

• The graphs of trigonometric functions exhibit periodic behavior
• The sine and cosine functions have a period of 2π
• The tangent function has a period of π
• The amplitude of sine and cosine functions is the maximum displacement from the x-axis
• Transformations such as shifts, stretches, and reflections can be applied to trigonometric graphs

Inverse Trigonometric Functions

• Inverse trigonometric functions find the angle whose trigonometric value is a given number
• The inverse sine function (arcsin or sin⁻¹) returns the angle whose sine is x
• The inverse cosine function (arccos or cos⁻¹) returns the angle whose cosine is x
• The inverse tangent function (arctan or tan⁻¹) returns the angle whose tangent is x
• The domains of inverse trigonometric functions are restricted to ensure they are one-to-one
• The ranges of arcsin, arccos, and arctan are [-π/2, π/2], [0, π], and (-π/2, π/2), respectively