Find the remaining five trigonometric functions of θ.
sin θ = -4/5, cos θ < 0

Verified step by step guidance

1. Identify the given information: \(\sin \theta = -\frac{4}{5}\) and \(\cos \theta < 0\). This tells us the sine value and the quadrant where \(\theta\) lies. Since sine is negative and cosine is negative, \(\theta\) is in the third quadrant.
2. Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Use this to find \(\cos \theta\) by substituting the known sine value: \(\left(-\frac{4}{5}\right)^2 + \cos^2 \theta = 1\).
3. Calculate \(\cos^2 \theta\) from the equation: \(\cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2\). Then take the square root to find \(\cos \theta\). Remember to choose the negative root because \(\cos \theta < 0\) in the third quadrant.
4. Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) you have found.
5. Calculate the remaining three trigonometric functions using their reciprocal relationships: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Substitute the known values to express each function.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given sin θ, this identity allows you to find cos θ by rearranging the equation. It is fundamental for determining unknown trigonometric functions when one function value is known.

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and other trig functions depends on the quadrant where the angle θ lies. Since sin θ = -4/5 (negative) and cos θ < 0 (also negative), θ is in the third quadrant, where both sine and cosine are negative. This helps determine the correct sign of the functions.

Definitions of the Six Trigonometric Functions

The six trig functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of sides in a right triangle or coordinates on the unit circle. Knowing sin θ and cos θ allows calculation of the others using their definitions, such as tan θ = sin θ / cos θ and sec θ = 1 / cos θ.