Precalculus: Trigonometric Functions, Reference Angles, and Quadrant Analysis

Q1. What are the definitions of the six trigonometric functions for an angle θ in standard position, using the unit circle and right triangles?

Background

Topic: Trigonometric Functions and the Unit Circle

This question is about understanding how the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are defined for any angle θ, using a point (x, y) on the terminal side of θ and the distance r from the origin.

Key Terms and Formulas

• Standard Position: An angle whose vertex is at the origin and initial side is along the positive x-axis.

• Unit Circle: A circle with radius 1 centered at the origin.

• For any point (x, y) on the terminal side of θ, $r = \sqrt{x^2 + y^2}$.

The six trigonometric functions are defined as:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

On the unit circle ($r = 1$): $\sin \theta = y$, $\cos \theta = x$.

Step-by-Step Guidance

1. Start by identifying a point (x, y) on the terminal side of angle θ in standard position.

2. Calculate the distance from the origin to this point using $r = \sqrt{x^2 + y^2}$.

3. Use the definitions above to express each trigonometric function in terms of x, y, and r.

4. If the point lies on the unit circle, substitute $r = 1$ to simplify the expressions for sine and cosine.

Try solving on your own before revealing the answer!

Final Answer:

The six trigonometric functions for an angle θ in standard position, with a point (x, y) on its terminal side and $r = \sqrt{x^2 + y^2}$, are:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

On the unit circle ($r = 1$): $\sin \theta = y$, $\cos \theta = x$.

Q2. What is the reference angle for 155°?

Background

Topic: Reference Angles

This question tests your understanding of how to find the reference angle for a given angle in degrees. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis.

Key Terms and Formulas

• Reference Angle: The acute angle (less than 90°) formed by the terminal side of the given angle and the x-axis.

• For angles in Quadrant II: $\text{Reference angle} = 180^\circ - \theta$

Step-by-Step Guidance

1. Determine which quadrant 155° is in (hint: 90° < 155° < 180° means Quadrant II).

2. Recall the formula for the reference angle in Quadrant II: $\text{Reference angle} = 180^\circ - \theta$.

3. Set up the calculation using the given angle.

Try solving on your own before revealing the answer!

Final Answer:

The reference angle for 155° is $180^\circ - 155^\circ = 25^\circ$.

This is the acute angle between the terminal side of 155° and the x-axis.

Q3. Use a reference angle to find $\tan 2115^\circ$.

Background

Topic: Coterminal Angles and Reference Angles

This question is about reducing a large angle to an equivalent angle between 0° and 360°, then using the reference angle to evaluate the tangent function.

Key Terms and Formulas

• Coterminal Angles: Angles that share the same terminal side, found by adding or subtracting multiples of 360°.

• Reference Angle: The acute angle between the terminal side and the x-axis.

• $\tan(\theta) = \tan(\theta + 360^\circ k)$ for any integer k.

Step-by-Step Guidance

1. Divide 2115° by 360° to find how many full rotations are completed and the remainder.

2. The remainder gives the coterminal angle between 0° and 360°.

3. Identify the reference angle for this coterminal angle.

4. Recall the sign of the tangent function in the relevant quadrant.

Try solving on your own before revealing the answer!

Final Answer:

2115° ÷ 360° = 5 remainder 315°, so $\tan 2115^\circ = \tan 315^\circ$.

The reference angle for 315° is 45°, and since 315° is in Quadrant IV, $\tan 315^\circ = -1$.

Q4. Given $\sec \theta = -\frac{5}{4}$ and $\theta$ is in Quadrant III, find $\sin \theta$ and $\tan \theta$.

Background

Topic: Trigonometric Functions in Different Quadrants

This question tests your ability to use the value of one trigonometric function and quadrant information to find the values of other trigonometric functions.

Key Terms and Formulas

• $\sec \theta = \frac{1}{\cos \theta}$

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$

• Quadrant III: Both sine and cosine are negative, tangent is positive.

Step-by-Step Guidance

1. Start by finding $\cos \theta$ from $\sec \theta = -\frac{5}{4}$.

2. Use the Pythagorean identity to solve for $\sin \theta$ (remember the sign in Quadrant III).

3. Set up the calculation for $\tan \theta = \frac{\sin \theta}{\cos \theta}$, considering the signs in Quadrant III.

Try solving on your own before revealing the answer!

Final Answer:

$\sec \theta = -\frac{5}{4}$ means $\cos \theta = -\frac{4}{5}$.

In Quadrant III, $\sin \theta < 0$. Compute $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}$, so $\sin \theta = -\frac{3}{5}$.

Then $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3/5}{-4/5} = \frac{3}{4}$ (positive in Quadrant III).