Right Triangle Trigonometry

Study Notes

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.

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.

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).

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.