Trigonometric Ratios to Know for Geometry

Trigonometric ratios are key tools in Geometry, helping us understand the relationships between angles and sides in right triangles. Sine, cosine, and tangent allow us to solve for unknown lengths and angles, making geometry more manageable and practical.

1. Sine (sin)

   • Defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
   • Formula: sin(θ) = opposite/hypotenuse.
   • Used to find unknown side lengths or angles in right triangles.

2. Cosine (cos)

   • Defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
   • Formula: cos(θ) = adjacent/hypotenuse.
   • Essential for solving problems involving angles and distances in right triangles.

3. Tangent (tan)

   • Defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
   • Formula: tan(θ) = opposite/adjacent.
   • Useful for determining angles and side lengths when only two sides are known.

4. SOH CAH TOA mnemonic

   • A memory aid to remember the definitions of sine, cosine, and tangent.
   • SOH: Sine = Opposite/Hypotenuse.
   • CAH: Cosine = Adjacent/Hypotenuse.
   • TOA: Tangent = Opposite/Adjacent.

5. Right triangle relationships

   • The three sides of a right triangle are related through trigonometric ratios.
   • The angles in a right triangle sum up to 90 degrees (excluding the right angle).
   • Trigonometric ratios remain consistent for a given angle across all right triangles.

6. Unit circle

   • A circle with a radius of 1 centered at the origin of a coordinate plane.
   • Used to define trigonometric functions for all angles, not just those in right triangles.
   • Coordinates on the unit circle correspond to the cosine and sine values of angles.

7. Reciprocal ratios (cosecant, secant, cotangent)

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

8. Angle of elevation and depression

   • Angle of elevation: the angle formed by a horizontal line and the line of sight to an object above the horizontal.
   • Angle of depression: the angle formed by a horizontal line and the line of sight to an object below the horizontal.
   • Both angles can be solved using trigonometric ratios in right triangles.

9. Special right triangles (30-60-90 and 45-45-90)

   • 30-60-90 triangle: sides are in the ratio 1:√3:2.
   • 45-45-90 triangle: sides are in the ratio 1:1:√2.
   • These ratios simplify calculations for angles and side lengths in these specific triangles.

10. Pythagorean theorem and its relation to trigonometric ratios

    • States that in a right triangle, a² + b² = c², where c is the hypotenuse.
    • Trigonometric ratios can be derived from the Pythagorean theorem.
    • Helps in finding missing side lengths when using sine, cosine, or tangent.