Trigonometry Cheat Sheet

Trigonometry explores the relationship between the sides and angles of triangles, with applications in navigation, surveying, and construction. It utilizes primary trigonometric ratios (SOH CAH TOA), the sine law, and the cosine law to solve for unknown sides and angles.

Core Principles

• Primary Trigonometric Ratios (SOH CAH TOA): sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
• Sine Law: Relates the ratio of the sine of an angle to the length of its opposite side in any triangle (a/sin A = b/sin B = c/sin C).
• Cosine Law: Relates the lengths of the three sides of a triangle to the cosine of an angle (a² = b² + c² - 2bc cos A).
• Solving Triangles: Determining all unknown angles and sides of a triangle using trigonometric laws and the Pythagorean theorem.
• Angles of Elevation and Depression: Angles measured upwards or downwards from the horizontal, respectively, used in real-world problem-solving.
• Applications: Used in fields like piloting, surveying, construction, physics, and engineering to calculate distances, heights, and angles.

Action Steps

• Identify the knowns and unknowns in the problem.
• Draw a diagram to model the situation, labeling all known sides and angles.
• Determine if the triangle is a right triangle to use primary trigonometric ratios.
• If not a right triangle, determine if the Sine Law or Cosine Law is appropriate based on the given information (e.g., SAS for Cosine Law, ASA/AAS for Sine Law).
• Set up the appropriate trigonometric equation.
• Solve for the unknown side or angle.
• Check the answer for reasonableness and units.

Formulas

• $ \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} $
• $ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} $
• $ \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} $
• $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $
• $ a^2 = b^2 + c^2 - 2bc \cos A $
• $ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $

Key Terms

• Adjacent: The side of a right triangle that is next to the angle and not the hypotenuse.
• Opposite: The side of a right triangle that is across from the angle.
• Hypotenuse: The longest side of a right triangle, opposite the right angle.
• Angle of Elevation: The angle between the horizontal and the line of sight to an object above the horizontal.
• Angle of Depression: The angle between the horizontal and the line of sight to an object below the horizontal.
• Complementary Angles: Two angles whose sum is 90°.
• Sine Law: Relates the ratio of the sine of an angle to the length of its opposite side in any triangle.
• Cosine Law: Relates the lengths of the three sides of a triangle to the cosine of an angle.
• SOH CAH TOA: An acronym to remember the primary trigonometric ratios: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), Tangent (Opposite/Adjacent).

Real World Examples

• A pilot navigating an aircraft.: Uses trigonometry to determine headings and distances based on angles and known distances.
• A surveyor measuring land.: Uses trigonometry to calculate distances and angles for property boundaries and terrain mapping.
• A carpenter building a roof truss.: Uses trigonometry to determine the angles and lengths of rafters for precise cuts and structural integrity.

People

• Demar: A geomatics student training to be a licensed surveyor, using trigonometry for land measurements.