Right Triangle Trigonometry

Right Triangle Trigonometry

• Trigonometric Functions: Six trigonometric functions are defined for an acute angle θ in a right triangle. These functions relate the ratios of the sides of the triangle to the angle.

  • sin θ = (opposite side)/(hypotenuse)
  • cos θ = (adjacent side)/(hypotenuse)
  • tan θ = (opposite side)/(adjacent side)
  • csc θ = (hypotenuse)/(opposite side)
  • sec θ = (hypotenuse)/(adjacent side)
  • cot θ = (adjacent side)/(opposite side)

• Example 1: Given a right triangle with hypotenuse 4 and adjacent side 3, the six trigonometric values of the angle θ can be found.

Evaluating Trigonometric Functions

• Example 1 (continued): Using the triangle (hypotenuse = 4, adjacent side = 3), these calculations can be performed:

  • sin θ = 3/4
  • cos θ = 4/3
  • tan θ = 3/4
  • csc θ = 4/3
  • sec θ = 3/4
  • cot θ = 4/3

• Example 2: Calculate the values of sin 45°, cos 45°, and tan 45° using the given right triangle. Specific values can be determined, such as for 45-45-90 triangles.

• Example 3: Calculate sin 60°, cos 60°, sin 30°, and cos 30° using an equilateral triangle. This example emphasizes triangles with specific angle measurements.

Trigonometric Identities

• Reciprocal Identities: These identities show the relationships between reciprocal trigonometric functions.

  • sin θ = 1/csc θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ
  • csc θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ

• Quotient Identities: These identities relate tangent and cotangent to sine and cosine.

  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ

• Pythagorean Identities: These relate sine and cosine, or tangent, cotangent, secant and cosecant to one another.

  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = csc² θ

• Example 5: Apply trigonometric identities to determine the value of sin θ (or another trigonometric ratio) given cos θ=0.8.

• Example 6: Apply trigonometric identities to determine the value of cot θ (or another trigonometric ratio) given tan θ=3.

• Example 7: Transform one side of trigonometric equations into the other by using trigonometric identities within a specified domain (range of θ values).

Solving Right Triangles Using Trigonometry

• Example 8: A surveyor measures the angle of elevation to the top of a tree. Trigonometry is used to find the height of the tree given the distance from the base of the tree and the angle of elevation.

• Example 9: A person is walking along a path that forms a specific angle with a riverbank. The given distance and angle to the river are used to determine the angle between the path and the river.

• Example 10: Find the length (hypotenuse) of a skateboard ramp, given its height and the angle of elevation.