Right triangle trigonometry is all about ratios in triangles with a 90-degree angle. It introduces sine, cosine, and tangent functions, which relate the sides of these triangles. These concepts are crucial for solving real-world problems involving heights, distances, and angles.

Understanding right triangle trig opens doors to more advanced math. It's the foundation for working with any angle, not just those in right triangles. This knowledge is key for tackling complex problems in physics, engineering, and other fields that use angular measurements.

Right Triangle Trigonometry

Trigonometric functions in right triangles

• Sine (sin) represents the ratio of the opposite side to the hypotenuse in a right triangle
• Cosine (cos) represents the ratio of the adjacent side to the hypotenuse in a right triangle
• Tangent (tan) represents the ratio of the opposite side to the adjacent side in a right triangle
• Reciprocal functions include cosecant (csc), secant (sec), and cotangent (cot)
  • Cosecant (csc) is the reciprocal of sine, calculated as the hypotenuse divided by the opposite side
  • Secant (sec) is the reciprocal of cosine, calculated as the hypotenuse divided by the adjacent side
  • Cotangent (cot) is the reciprocal of tangent, calculated as the adjacent side divided by the opposite side
• To evaluate trigonometric functions in a right triangle:
  1. Identify the angle of interest within the triangle
  2. Label the sides of the right triangle as opposite, adjacent, and hypotenuse relative to the angle of interest
  3. Use the appropriate trigonometric function ratio (sin, cos, tan, csc, sec, or cot) to calculate the desired value based on the given information
• Inverse trigonometric functions (e.g., arcsin, arccos, arctan) can be used to find angles when given side ratios

Common angle trigonometric values

• Special right triangles have specific side length ratios that simplify trigonometric calculations
  • 30°-60°-90° triangle has side lengths in the ratio of 1 : $\sqrt{3}$ : 2, with the hypotenuse being twice the length of the shortest side
  • 45°-45°-90° triangle has side lengths in the ratio of 1 : 1 : $\sqrt{2}$, with the hypotenuse being $\sqrt{2}$ times the length of a leg
• Trigonometric function values for common angles:
  • 30°: sin = 1/2, cos = $\sqrt{3}/2$, tan = $\sqrt{3}/3$
  • 45°: sin = $\sqrt{2}/2$, cos = $\sqrt{2}/2$, tan = 1
  • 60°: sin = $\sqrt{3}/2$, cos = 1/2, tan = $\sqrt{3}$
• Memorizing these common angle values can help simplify trigonometric calculations and serve as reference points for estimating function values of other angles

Cofunctions and complementary angles

• Cofunctions are pairs of trigonometric functions that are related by the complement of an angle (90° - $\theta$)
  • Cofunction pairs include sin and cos, tan and cot, sec and csc
  • Cofunction identities express the relationship between a function and its cofunction, such as $\sin(\theta) = \cos(90° - \theta)$ and $\cos(\theta) = \sin(90° - \theta)$
• Complementary angles are two angles whose sum is 90°
  • To find the trigonometric function value of an angle, you can use the cofunction of its complement
  • For example, if $\theta = 30°$, then $\sin(30°) = \cos(60°)$ because 30° and 60° are complementary angles

Trigonometric functions for any angle

• Angle conventions define the direction and orientation of angles in the coordinate plane
  • Positive angles are measured counterclockwise from the positive x-axis
  • Negative angles are measured clockwise from the positive x-axis
• The unit circle is a circle with a radius of 1 centered at the origin
  • Angle $\theta$ is formed by the positive x-axis and a line from the origin to a point (x, y) on the unit circle
  • Trigonometric functions can be defined using the unit circle, where $\sin(\theta) = y$, $\cos(\theta) = x$, and $\tan(\theta) = y/x$
• Trigonometric functions in the coordinate plane:
  • Sine represents the y-coordinate of the point on the unit circle corresponding to the angle
  • Cosine represents the x-coordinate of the point on the unit circle corresponding to the angle
  • Tangent represents the slope of the line from the origin to the point on the unit circle corresponding to the angle
• The reference angle is the acute angle formed between the terminal side of an angle and the x-axis, used to evaluate trigonometric functions in different quadrants

Applications of right triangle trigonometry

• Right triangle trigonometry has numerous real-world applications, such as:
  • Finding heights and distances of objects (buildings, trees) using trigonometric functions
  • Calculating angles of elevation (angle between the horizontal and the line of sight to an object above the observer) and angles of depression (angle between the horizontal and the line of sight to an object below the observer)
  • Solving problems in navigation (determining distances and directions) and surveying (measuring angles and distances to create maps or plans)
• Problem-solving steps for applying right triangle trigonometry:
  1. Sketch a diagram of the problem situation, clearly labeling the given information and the unknown value to be found
  2. Identify the known values (distances or angles) and the unknown value in the diagram
  3. Set up a trigonometric equation using the appropriate function (sin, cos, or tan) based on the relationship between the known and unknown values
  4. Solve the equation for the unknown value, using algebra and inverse trigonometric functions if necessary

Extended Trigonometric Concepts

• Quadrants: The coordinate plane is divided into four quadrants, each with specific signs for trigonometric functions
• Trigonometric identities: Mathematical relationships between trigonometric functions that are always true, used to simplify expressions and solve equations
• Periodic functions: Trigonometric functions repeat their values at regular intervals, with periods determined by their specific properties