Right Triangle Trigonometry

• Trigonometry explores the correlation between triangle side lengths and angles.
• Right triangles, characterized by a 90-degree angle, are a key application of trigonometry.
• Trigonometric functions establish a connection between a triangle's angles and the ratios of its sides.
• Sine (sin), cosine (cos), and tangent (tan) form the foundation of trigonometric functions.

Right Triangle Trigonometry

• Right triangles feature an angle denoted as θ.
• The side opposite θ is the "opposite" side.
• The "adjacent" side sits next to θ, excluding the hypotenuse.
• The "hypotenuse" is the longest side, opposite the right angle.
• The sine (sin) of θ equals the opposite side divided by the hypotenuse: sin(θ) = Opposite / Hypotenuse.
• The cosine (cos) of θ equals the adjacent side divided by the hypotenuse: cos(θ) = Adjacent / Hypotenuse.
• The tangent (tan) of θ equals the opposite side divided by the adjacent side: tan(θ) = Opposite / Adjacent.
• SOH CAH TOA is a mnemonic for remembering trigonometric ratios:
• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Reciprocal Trigonometric Functions

• Cosecant (csc) is the reciprocal of sine: csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite.
• Secant (sec) is the reciprocal of cosine: sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent.
• Cotangent (cot) is the reciprocal of tangent: cot(θ) = 1 / tan(θ) = Adjacent / Opposite.

Trigonometric Values for Special Angles

• Angles like 0°, 30°, 45°, 60°, and 90° are commonly used in trigonometry.
• sin(0°) = 0, cos(0°) = 1, tan(0°) = 0
• sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3
• 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 true for all variable values where the trigonometric functions are defined.
• Pythagorean Identity: sin²(θ) + cos²(θ) = 1
• This stems from the Pythagorean theorem (a² + b² = c²) in right triangles.
• Variations: sin²(θ) = 1 - cos²(θ) and cos²(θ) = 1 - sin²(θ)
• Quotient Identities:
• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)
• Reciprocal Identities:
• csc(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ)
• Angle Sum and Difference Identities:
• 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 Identities:
• 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))

Law of Sines

• The Law of Sines connects triangle side lengths to the sines of opposite angles.
• For any triangle ABC, where sides a, b, c are opposite angles A, B, C: a / sin(A) = b / sin(B) = c / sin(C)
• The Law of Sines helps solve triangles when you have:
• Two angles and one side (AAS or ASA).
• Two sides and an angle opposite one of them (SSA, ambiguous case).

Law of Cosines

• The Law of Cosines links side lengths to the cosine of one angle.
• For any triangle ABC, where sides a, b, c are opposite angles A, B, C:
• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)
• The Law of Cosines helps solve triangles when you have:
• Three sides (SSS).
• Two sides and the included angle (SAS).

Radians

• Radians are an alternative angle measurement based on circle radius.
• A full circle (360°) is 2π radians.
• A straight angle (180°) is π radians.
• Degrees convert to radians by multiplying by π/180.
• Radians convert to degrees by multiplying by 180/π.
• The arc length (s) of a circle, with radius r, subtended by angle θ (in radians) is: s = rθ.

Unit Circle

• The unit circle has a radius of 1, centered at (0,0) on the Cartesian plane.
• Angles increase counterclockwise from the positive x-axis.
• For any point (x, y) on the unit circle at angle θ:
• x = cos(θ)
• y = sin(θ)
• The unit circle aids in visualizing trigonometric functions and their values.
• It clarifies the signs of trigonometric functions across different quadrants.

Graphing Trigonometric Functions

• y = sin(x) has a period of 2π, an amplitude of 1, and oscillates between -1 and 1.
• y = cos(x) has a period of 2π, an amplitude of 1, oscillating between -1 and 1; it's a horizontal shift of y = sin(x).
• y = tan(x) has a period of π and vertical asymptotes where cos(x) = 0.
• General forms of sine and cosine functions:
• y = A * sin(B(x - C)) + D
• y = A * cos(B(x - C)) + D
• A is the amplitude.
• B affects the period: Period = 2π / |B|.
• C is the horizontal (phase) shift.
• D is the vertical shift.

Inverse Trigonometric Functions

• Inverse trigonometric functions reverse the standard trigonometric functions.
• arcsin(x) or sin⁻¹(x) inverts sine, yielding values between [-π/2, π/2].
• arccos(x) or cos⁻¹(x) inverts cosine, yielding values between [0, π].
• arctan(x) or tan⁻¹(x) inverts tangent, yielding values between (-π/2, π/2).
• These determine angles from side ratios.