Question

Define the six trigonometric functions in terms of the sides of a right triangle.

Short answer

Question: Define the six trigonometric functions in terms of the sides of a right triangle. Answer: The six trigonometric functions are defined as follows: 1. sin(θ) = Opposite / Hypotenuse 2. cos(θ) = Adjacent / Hypotenuse 3. tan(θ) = Opposite / Adjacent 4. csc(θ) = Hypotenuse / Opposite 5. sec(θ) = Hypotenuse / Adjacent 6. cot(θ) = Adjacent / Opposite

Step-by-step solution

Recall the SOHCAHTOA mnemonic

To define the six trigonometric functions, start by recalling the mnemonic SOHCAHTOA. SOHCAHTOA stands for: - Sine (sin) = Opposite / Hypotenuse - Cosine (cos) = Adjacent / Hypotenuse - Tangent (tan) = Opposite / Adjacent These functions are the primary trigonometric functions and we will use them to define the remaining three, which are the reciprocal functions.

Define the sine function and its reciprocal, the cosecant function

The sine function, abbreviated as sin, can be defined as the ratio of the length of the side opposite the angle (O) to the length of the hypotenuse (H). In a right triangle with an angle θ, the sine function is: sin(θ) = O/H The reciprocal function of sine is the cosecant function (csc): csc(θ) = 1/sin(θ) = H/O

Define the cosine function and its reciprocal, the secant function

The cosine function, abbreviated as cos, can be defined as the ratio of the length of the side adjacent to the angle (A) to the length of the hypotenuse (H). In a right triangle with an angle θ, the cosine function is: cos(θ) = A/H The reciprocal function of cosine is the secant function (sec): sec(θ) = 1/cos(θ) = H/A

Define the tangent function and its reciprocal, the cotangent function

The tangent function, abbreviated as tan, can be defined as the ratio of the length of the side opposite the angle (O) to the length of the side adjacent to the angle (A). In a right triangle with an angle θ, the tangent function is: tan(θ) = O/A The reciprocal function of tangent is the cotangent function (cot): cot(θ) = 1/tan(θ) = A/O Now we have defined all six trigonometric functions in terms of the sides of a right triangle.