Trigonometric ratios are the building blocks of right triangle math. They connect side lengths to angles, helping us solve real-world problems. Understanding sine, cosine, and tangent is key to mastering these relationships.

Special angles and their ratios pop up everywhere in math and science. Knowing these values by heart, along with reciprocal functions, gives you a powerful toolkit for tackling complex problems in geometry, physics, and engineering.

Trigonometric Ratios and Right Triangles

Trigonometric ratios in right triangles

• Sine (sin) of an angle relates opposite side to hypotenuse $sin θ = \frac{opposite}{hypotenuse}$ measures vertical component of angle
• Cosine (cos) of an angle relates adjacent side to hypotenuse $cos θ = \frac{adjacent}{hypotenuse}$ measures horizontal component of angle
• Tangent (tan) of an angle relates opposite side to adjacent side $tan θ = \frac{opposite}{adjacent}$ measures steepness of angle
• Mnemonic device SOH-CAH-TOA aids memory of ratios (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent)

Side lengths vs angle measures

• Pythagorean theorem $a^2 + b^2 = c^2$ relates side lengths in right triangles (c is hypotenuse)
• Angle sum theorem states all angles in a triangle always total 180° (triangle interior angle sum)
• Complementary angles in right triangles two non-right angles sum to 90° (complementary pair)
• Angle-side relationships longest side (hypotenuse) opposes right angle second-longest side opposes second-largest angle

Values for special angles

• 30-60-90 triangle side ratios 1 : √3 : 2 common in geometry and trigonometry
  • $sin 30° = \frac{1}{2}$, $cos 30° = \frac{\sqrt{3}}{2}$, $tan 30° = \frac{\sqrt{3}}{3}$ frequently used in physics
  • $sin 60° = \frac{\sqrt{3}}{2}$, $cos 60° = \frac{1}{2}$, $tan 60° = \sqrt{3}$ important in engineering calculations
• 45-45-90 triangle side ratios 1 : 1 : √2 isosceles right triangle
  • $sin 45° = cos 45° = \frac{\sqrt{2}}{2}$, $tan 45° = 1$ useful in computer graphics and design

Reciprocal trigonometric ratios

• Cosecant (csc) reciprocal of sine $csc θ = \frac{1}{sin θ} = \frac{hypotenuse}{opposite}$ used in advanced trigonometry
• Secant (sec) reciprocal of cosine $sec θ = \frac{1}{cos θ} = \frac{hypotenuse}{adjacent}$ applied in calculus and physics
• Cotangent (cot) reciprocal of tangent $cot θ = \frac{1}{tan θ} = \frac{adjacent}{opposite}$ useful in engineering problems
• Relationships between reciprocal functions demonstrate inverse nature
  • $sin θ csc θ = 1$
  • $cos θ sec θ = 1$
  • $tan θ cot θ = 1$