Problem 32

Question

find the exact value of each of the remaining trigonometric functions of $\theta$ $$ \tan \theta=\frac{5}{12}, \quad \cos \theta<0 $$

Step-by-Step Solution

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Answer

Sine, cosine, cotangent, cosecant, and secant of $\theta$ are $\frac{5}{13}$, $-\frac{12}{13}$, $-\frac{12}{5}$, $\frac{13}{5}$, and $-\frac{13}{12}$ respectively.

1Step 1: Identify triangle sides

Since $\tan{\theta} = \frac{5}{12}$, we recognize this as 'opposite/adjacent'. Let's form a right triangle, let the opposite side be 5, the adjacent be 12. In the triangle, use Pythagorean Theorem to find the hypotenuse $h$. The Pythagorean Theorem states $a^2 + b^2 = h^2$, where a and b are the two sides of the triangle. So, the hypotenuse is $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$.

2Step 2: Determine quadrant

Cosine is negative which means $\theta$ is either in the second or third quadrant. However, tangent is positive only in the first and third quadrants. Because cosine is negative and tangent is positive, $\theta$ must be in the second quadrant. In the second quadrant, sine is positive, but cosine, tangent and their reciprocals are negative.

3Step 3: Compute remaining trigonometric functions

Now, calculate the remaining functions using the right triangle created. 1) $\sin{\theta} = \frac{opposite}{hypotenuse} = \frac{5}{13}$2) $\cos{\theta} = \frac{adjacent}{hypotenuse} = \frac{12}{13}$. But since $\theta$ is in quadrant 2, $\cos{\theta}$ is negative: $-\frac{12}{13}$3) $\cot{\theta} = \frac{1}{\tan{\theta}} = \frac{12}{5}$ and is negative in the second quadrant, so $-\frac{12}{5}$. 4) $\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{13}{5}$5) $\sec{\theta} = \frac{1}{\cos{\theta}} = -\frac{13}{12}$ as $\cos{\theta}$ is negative.

Key Concepts

Tangent FunctionPythagorean TheoremQuadrants in TrigonometrySine FunctionCosine Function

Tangent Function

The tangent function is one of the primary trigonometric functions, representing the ratio of the opposite side to the adjacent side in a right triangle. When dealing with an angle $ \theta $, the tangent value $ \tan{\theta} $ can be expressed as:

$ \tan{\theta} = \frac{\text{opposite}}{\text{adjacent}} $.

For this exercise, the tangent function is given as $ \tan{\theta} = \frac{5}{12} $. This means the side opposite the angle $ \theta $ is 5 units, while the adjacent side is 12 units.
Understanding this ratio helps in creating a right triangle which can be used to find other trigonometric functions via the Pythagorean theorem.
Additionally, remember that the tangent function has unique behaviors in different quadrants of the coordinate plane, influencing its sign based on which quadrant $ \theta $ lies in.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, extremely useful when analyzing right-angled triangles. It is expressed as:

$ a^2 + b^2 = c^2 $

where $ a $ and $ b $ represent the lengths of the two shorter sides, and $ c $ is the hypotenuse of the triangle.
In the context of trigonometry, once you know two sides of a right triangle, you can determine the third. Given $ \tan{\theta} = \frac{5}{12} $, use the theorem to find the hypotenuse:

$ 5^2 + 12^2 = h^2 $
$ 25 + 144 = h^2 $
$ h = \sqrt{169} = 13 $

With this hypotenuse, you can calculate sine and cosine values for angle $ \theta $.

Quadrants in Trigonometry

Trigonometry divides the coordinate plane into four quadrants, each affecting the sign of the six trigonometric functions. The placement of angle $ \theta $ in these quadrants determines whether values like sine, cosine, and tangent are positive or negative:

First Quadrant: All trigonometric functions are positive.
Second Quadrant: Sine is positive; cosine and tangent are negative.
Third Quadrant: Tangent is positive; sine and cosine are negative.
Fourth Quadrant: Cosine is positive; sine and tangent are negative.

In our problem, we're told $ \cos{\theta} < 0 $ and $ \tan{\theta} > 0 $. This combination of signs tells us that $ \theta $ is in the third quadrant.

Sine Function

The sine function relates to the ratio of the opposite side over the hypotenuse in a right triangle. It is expressed as:

$ \sin{\theta} = \frac{\text{opposite}}{\text{hypotenuse}} $

In our scenario with a hypotenuse calculated as 13 using the Pythagorean theorem, and the opposite side as 5, we find:

$ \sin{\theta} = \frac{5}{13} $

Given that $ \theta $ lies in the second quadrant, the sine function remains positive, validating that $ \sin{\theta} = \frac{5}{13} $. The sine value is crucial for calculating the cosecant, which is the reciprocal of the sine function.