Question

A point on the terminal side of an angle \(\theta\) in standard position is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ \left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right) $$

Short answer

\(\sin(\theta) = \cos(\theta) = -\frac{\sqrt{2}}{2}, \tan(\theta) = 1, \csc(\theta) = \sec(\theta) = -\sqrt{2}, \cot(\theta) = 1\).

Step-by-step solution

- Identify Coordinates and Quadrant

The terminal side of \(\theta\) passes through the point \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\. Both coordinates are negative, indicating that the point is in the third quadrant.

- Calculate radius \(r\)

Using the coordinates, calculate \(r\) using the Pythagorean theorem: \[ r = \sqrt{\left( -\frac{\sqrt{2}}{2} \right)^2 + \left( -\frac{\sqrt{2}}{2} \right)^2 } = \sqrt{\frac{2}{4} + \frac{2}{4}} = \sqrt{1} = 1 \].

- Find Sine and Cosine

Using the definitions of sine and cosine, we get: \[ \sin(\theta) = \frac{y}{r} = \-\frac{\sqrt{2}}{2}, \quad \cos(\theta) = \frac{x}{r} = \-\frac{\sqrt{2}}{2} \].

- Find Tangent

Tangent is the ratio of sine to cosine: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \].

- Find Cosecant

Cosecant is the reciprocal of sine: \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \].

- Find Secant

Secant is the reciprocal of cosine: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \].

- Find Cotangent

Cotangent is the reciprocal of tangent: \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{1} = 1 \].

Key concepts

trigonometric identities

Trigonometric identities are fundamental in solving problems involving trigonometric functions. They are equations that hold true for all values of the variable within their domains. Important identities include: the Pythagorean identities like \[ \sin^2(\theta) + \cos^2(\theta) = 1 \], quotient identities like \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \], and reciprocal identities such as \[ \csc(\theta) = \frac{1}{\sin(\theta)} \] and \[ \sec(\theta) = \frac{1}{\cos(\theta)} \]. These identities are immensely helpful for simplifying expressions and solving equations in trigonometry. Understanding these identities can also aid in recognizing patterns and making connections between different trigonometric functions.

unit circle

The unit circle is a crucial concept in trigonometry because it provides a geometric representation of the trigonometric functions. It is a circle with a radius of 1, centered at the origin of the coordinate system. Points on the unit circle correspond to angles measured from the positive x-axis. For any angle \( \theta \), the coordinates \[ (x, y) \] on the unit circle are \[ \cos(\theta) \] and \[ \sin(\theta) \], respectively. This makes it easy to visualize and calculate the values of trigonometric functions. For example, the point \[ \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \] falls on the unit circle and relates to certain angles, providing insight into the trigonometric values at these angles.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that is also widely used in trigonometry. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as \[ a^2 + b^2 = c^2 \]. In the context of trigonometric functions, the Pythagorean theorem helps calculate the radius \( r \) of a point on the terminal side of an angle \( \theta \). Given coordinates \( (x, y) \), the radius can be found using \[ r = \sqrt{x^2 + y^2} \]. For example, for the point \[ \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \], the calculation \[ r = \sqrt{ \left( -\frac{\sqrt{2}}{2} \right)^2 + \left( -\frac{\sqrt{2}}{2} \right)^2 } = 1 \] follows from this theorem.

reciprocal trigonometric functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions by taking their reciprocals. They are crucial for solving many trigonometric problems. The three main reciprocal functions are cosecant \[ \csc(\theta) = \frac{1}{\sin(\theta)} \], secant \[ \sec(\theta) = \frac{1}{\cos(\theta)} \], and cotangent \[ \cot(\theta) = \frac{1}{\tan(\theta)} \]. For example, in solving our original problem, we found that \[ \sin(\theta) = -\frac{\sqrt{2}}{2} \], thus \[ \csc(\theta) = -\sqrt{2} \]. Similarly, with \[ \cos(\theta) = -\frac{\sqrt{2}}{2} \], we have \[ \sec(\theta) = -\sqrt{2} \]. Understanding reciprocal functions helps in writing and manipulating trigonometric equations in diverse forms.