Finding Exact Trigonometric Values Given Cosine And Tangent

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In trigonometry, the six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—are fundamental for understanding angles and their relationships to the sides of a right triangle. When the value of one trigonometric function is known, along with the quadrant in which the angle lies, it is possible to determine the values of the other trigonometric functions. This process involves utilizing trigonometric identities and the definitions of the functions in terms of the sides of a right triangle: the opposite, adjacent, and hypotenuse.

This article provides a detailed guide on how to find the exact values of the remaining trigonometric functions when given the values of cosine and tangent. We will delve into the necessary steps and explanations, ensuring that you grasp the underlying principles and can apply them to various problems. Understanding these relationships is crucial not only in mathematics but also in various fields such as physics, engineering, and computer graphics.

Before diving into the specifics, let's briefly review the definitions of the six trigonometric functions in the context of a right triangle. Consider a right triangle with an angle θ. The sides are defined as follows:

  • Opposite: The side opposite to angle θ.
  • Adjacent: The side adjacent to angle θ.
  • Hypotenuse: The longest side, opposite the right angle.

Given these sides, the trigonometric functions are defined as:

  • Sine (sin θ): The ratio of the opposite side to the hypotenuse (sin θ = Opposite / Hypotenuse).
  • Cosine (cos θ): The ratio of the adjacent side to the hypotenuse (cos θ = Adjacent / Hypotenuse).
  • Tangent (tan θ): The ratio of the opposite side to the adjacent side (tan θ = Opposite / Adjacent).
  • Cosecant (csc θ): The reciprocal of sine (csc θ = Hypotenuse / Opposite).
  • Secant (sec θ): The reciprocal of cosine (sec θ = Hypotenuse / Adjacent).
  • Cotangent (cot θ): The reciprocal of tangent (cot θ = Adjacent / Opposite).

These definitions form the basis for understanding and calculating trigonometric values. Additionally, it is important to remember the signs of these functions in different quadrants of the coordinate plane. The mnemonic "All Students Take Calculus" can be helpful:

  • Quadrant I (All): All trigonometric functions are positive.
  • Quadrant II (Students): Sine and cosecant are positive.
  • Quadrant III (Take): Tangent and cotangent are positive.
  • Quadrant IV (Calculus): Cosine and secant are positive.

Consider the scenario where we are given the values of the cosine and tangent functions for an angle θ. Specifically, we have:

cos⁡θ=−437\cos \theta = -\frac{4\sqrt{3}}{7}

tan⁡θ=−312\tan \theta = -\frac{\sqrt{3}}{12}

We also know that:

cot⁡θ=−43\cot \theta = -4\sqrt{3}

The task is to find the exact values of the remaining trigonometric functions, which are secant (sec θ) and cosecant (csc θ). This involves using the given values and the fundamental trigonometric identities to derive the unknown values.

1. Determine the Quadrant

First, we need to determine which quadrant the angle θ lies in. This can be done by analyzing the signs of the given trigonometric functions:

  • cos⁡θ\cos \theta is negative.
  • tan⁡θ\tan \theta is negative.

Cosine is negative in the second and third quadrants, while tangent is negative in the second and fourth quadrants. The common quadrant where both cosine and tangent are negative is the second quadrant. Therefore, angle θ lies in the second quadrant. This information is crucial as it helps us determine the signs of the remaining trigonometric functions.

2. Find Secant (sec θ)

The secant function is the reciprocal of the cosine function. Therefore, to find sec θ, we simply take the reciprocal of the given cos θ value:

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

Given that cos⁡θ=−437\cos \theta = -\frac{4\sqrt{3}}{7}, we have:

sec⁡θ=1−437=−743\sec \theta = \frac{1}{-\frac{4\sqrt{3}}{7}} = -\frac{7}{4\sqrt{3}}

To rationalize the denominator, we multiply both the numerator and the denominator by 3\sqrt{3}:

sec⁡θ=−743⋅33=−734⋅3=−7312\sec \theta = -\frac{7}{4\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{7\sqrt{3}}{4 \cdot 3} = -\frac{7\sqrt{3}}{12}

Thus, the exact value of the secant function is:

sec⁡θ=−7312\sec \theta = -\frac{7\sqrt{3}}{12}

Since we are in the second quadrant, where cosine and secant are negative, this value aligns with our expectations.

3. Find Sine (sin θ)

To find the sine function, we can use the identity that relates sine, cosine, and tangent:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

We are given tan⁡θ=−312\tan \theta = -\frac{\sqrt{3}}{12} and cos⁡θ=−437\cos \theta = -\frac{4\sqrt{3}}{7}. Rearranging the identity to solve for sin θ, we get:

sin⁡θ=tan⁡θ⋅cos⁡θ\sin \theta = \tan \theta \cdot \cos \theta

Substituting the given values:

sin⁡θ=(−312)⋅(−437)\sin \theta = \left(-\frac{\sqrt{3}}{12}\right) \cdot \left(-\frac{4\sqrt{3}}{7}\right)

Multiplying the fractions:

sin⁡θ=3⋅4312⋅7=4⋅312⋅7=1284\sin \theta = \frac{\sqrt{3} \cdot 4\sqrt{3}}{12 \cdot 7} = \frac{4 \cdot 3}{12 \cdot 7} = \frac{12}{84}

Simplifying the fraction:

sin⁡θ=17\sin \theta = \frac{1}{7}

Therefore, the exact value of the sine function is:

sin⁡θ=17\sin \theta = \frac{1}{7}

In the second quadrant, sine is positive, which is consistent with our result.

4. Find Cosecant (csc θ)

The cosecant function is the reciprocal of the sine function. Therefore, to find csc θ, we take the reciprocal of the sin θ value:

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Given that sin⁡θ=17\sin \theta = \frac{1}{7}, we have:

csc⁡θ=117=7\csc \theta = \frac{1}{\frac{1}{7}} = 7

Thus, the exact value of the cosecant function is:

csc⁡θ=7\csc \theta = 7

Since we are in the second quadrant, where sine and cosecant are positive, this value is also consistent with our expectations.

Given cos⁡θ=−437\cos \theta = -\frac{4\sqrt{3}}{7} and tan⁡θ=−312\tan \theta = -\frac{\sqrt{3}}{12}, we have found the following values for the remaining trigonometric functions:

  • Secant (sec θ): sec⁡θ=−7312\sec \theta = -\frac{7\sqrt{3}}{12}
  • Sine (sin θ): sin⁡θ=17\sin \theta = \frac{1}{7}
  • Cosecant (csc θ): csc⁡θ=7\csc \theta = 7

These results provide a complete set of trigonometric function values for the angle θ, satisfying the given conditions and adhering to the properties of the second quadrant.

Determining the exact values of trigonometric functions when given certain conditions, such as the values of cosine and tangent, requires a strong understanding of trigonometric identities and the properties of trigonometric functions in different quadrants. By following a systematic approach, we can accurately calculate the values of the remaining functions.

In this article, we successfully found the values of secant and cosecant given the values of cosine and tangent. The key steps included identifying the quadrant, using reciprocal identities, and applying the relationship between sine, cosine, and tangent. These methods are crucial for solving a variety of trigonometric problems and are fundamental in many areas of mathematics and its applications.

By understanding and practicing these techniques, you can confidently tackle trigonometric problems and gain a deeper appreciation for the relationships between trigonometric functions and angles.