It has been a while since I last delved into trigonometry, but recently a problem arose that needed a review of some basic trigonometric identities and their relationships. Specifically, we were given that csc(θ) 3.87, and we needed to find the values of the other five trigonometric functions for the angle θ. Let's break this down step by step.

Understanding the Problem

The Importance of Cosecant in Trigonometry

csc(θ) is the coterminal with sin(θ), meaning that csc(θ) 1 / sin(θ). If we were to simply solve for sin(θ) using the definition of cosecant, we would get:

csc(θ) 3.87 implies sin(θ) 1 / 3.87 0.258

Determining the Angle θ

Using the sine value, we can now determine the angle θ. The angle θ corresponding to sin(θ) 0.258 can be found using a calculator or by recognizing an approximate angle. Generally, θ ≈ 15°, since sine values around 15° are in that range.

Calculating the Remaining Trigonometric Functions

Sine and Cosine

Now that we have sin(θ) 0.258 and the angle θ ≈ 15°, we can use basic trigonometric identities to find the other functions.

cos(θ) 0.966 sec(θ) 1 / cos(θ) ≈ 1 / 0.966 ≈ 1.035 tan(θ) sin(θ) / cos(θ) ≈ 0.258 / 0.966 ≈ 0.268 cot(θ) 1 / tan(θ) ≈ 1 / 0.268 ≈ 3.723

Review of Trigonometric Functions and Important Triangles

Understanding the Definitions

To provide a clearer understanding, let's revisit the definitions of the trigonometric functions in terms of the sides of a right-angled triangle:

Sine (sin): The ratio of the length of the opposite side to the hypotenuse. Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse. Tangent (tan): The ratio of the length of the opposite side to the adjacent side. Cosecant (csc): The reciprocal of sine, i.e., csc(θ) 1 / sin(θ). Secant (sec): The reciprocal of cosine, i.e., sec(θ) 1 / cos(θ). Cotangent (cot): The reciprocal of tangent, i.e., cot(θ) 1 / tan(θ).

Memorizing Specific Right-Angled Triangles

One of the most useful right-angled triangles for memorization is the 3-4-5 triangle, which satisfies the Pythagorean theorem:

32 42 52

Knowing this, we can easily determine:

For a 15° angle, the sine is approximately 0.258. Using the 3-4-5 triangle, we can find the cosine and tangent values.

Practical Application in Trigonometry

Calculator Usage

Using a calculator is often the quickest way to verify these calculations. Most scientific calculators have built-in functions for sine, cosine, and tangent. Here is a guide on how to use these functions:

Sin: Press the SIN button, input the angle (15° in this case), and press equals. Cos: Press the COS button, input the angle (15° in this case), and press equals. Tan: Press the TAN button, input the angle (15° in this case), and press equals.

Demonstration with Calculator

For example, if you use a calculator, you would find:

sin(15°) 0.258794764 cos(15°) 0.965925826 tan(15°) 0.267949192

Final Summary

In summary, given that csc(θ) 3.87, we can calculate the other trigonometric functions as follows:

Trigonometric Values for θ ≈ 15°

sin(θ) ≈ 0.258 cos(θ) ≈ 0.966 tan(θ) ≈ 0.268 sec(θ) ≈ 1.035 cot(θ) ≈ 3.723

These values are approximations and depend on the precision of the calculator. Understanding these relationships is crucial for solving more complex trigonometric problems and for applying trigonometric functions in real-world scenarios.