Introduction to Trigonometry

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

Right-Angled Triangles

• Trigonometry is often introduced in the context of right-angled triangles
• A right-angled triangle has one angle that measures 90 degrees
• The side opposite the right angle is called 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
• The side across from the specified angle is the opposite
• The side next to the specified angle is the adjacent, it is not the hypotenuse

Trigonometric Ratios

• The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan)
• These ratios relate the angles of a right-angled triangle to the ratios of its sides
• Sine of an angle ((\theta)): (\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}})
• Cosine of an angle ((\theta)): (\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}})
• Tangent of an angle ((\theta)): (\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}})
• A common mnemonic to remember these ratios is SOH CAH TOA
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Reciprocal Trigonometric Ratios

• Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocal trigonometric ratios
• They are defined as follows:
  • Cosecant of an angle ((\theta)): (\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}})
  • Secant of an angle ((\theta)): (\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}})
  • Cotangent of an angle ((\theta)): (\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}})

Common Angles

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

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables
• Pythagorean Identity: (\sin^2(\theta) + \cos^2(\theta) = 1)
  • Variations: (\sin^2(\theta) = 1 - \cos^2(\theta)) and (\cos^2(\theta) = 1 - \sin^2(\theta))
• Quotient Identities:
  • (\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)})
  • (\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)})
• Reciprocal Identities:
  • (\csc(\theta) = \frac{1}{\sin(\theta)})
  • (\sec(\theta) = \frac{1}{\cos(\theta)})
  • (\cot(\theta) = \frac{1}{\tan(\theta)})

Angle Sum and Difference Formulas

• These formulas relate trigonometric functions of sums or differences of angles to trigonometric functions of the individual angles
• Sine Sum: (\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B))
• Sine Difference: (\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B))
• Cosine Sum: (\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B))
• Cosine Difference: (\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B))
• Tangent Sum: (\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)})
• Tangent Difference: (\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)})

Double-Angle Formulas

• These are special cases of the sum formulas where the two angles are equal
• Sine Double-Angle: (\sin(2\theta) = 2\sin(\theta)\cos(\theta))
• Cosine Double-Angle: (\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta))
• Tangent Double-Angle: (\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)})

Half-Angle Formulas

• These formulas express trigonometric functions of half an angle in terms of trigonometric functions of the full angle
• Sine Half-Angle: (\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}})
• Cosine Half-Angle: (\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}})
• Tangent Half-Angle: (\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)})
• The (\pm) sign depends on the quadrant in which (\frac{\theta}{2}) lies

Law of Sines

• Relates the lengths of the sides of a triangle to the sines of its angles
• Given a triangle with sides (a), (b), (c) and angles (A), (B), (C) opposite those sides respectively:
• (\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)})
• Used to find unknown sides or angles in a triangle when given certain information

Law of Cosines

• Relates the lengths of the sides of a triangle to the cosine of one of its angles
• Given a triangle with sides (a), (b), (c) and angle (C) opposite side (c):
• (c^2 = a^2 + b^2 - 2ab\cos(C))
• Can be rearranged to find the angle (C): (\cos(C) = \frac{a^2 + b^2 - c^2}{2ab})
• Used to find unknown sides or angles in a triangle when given certain information

Applications of Trigonometry

• Navigation: Determining directions and distances on maps
• Physics: Analyzing projectile motion, wave phenomena, and oscillations
• Engineering: Designing structures, bridges, and machines
• Surveying: Measuring land and creating maps
• Astronomy: Calculating distances to stars and planets