Precalculus Trigonometry and Functions Study Guide – Step-by-Step Guidance

Q1. Draw each angle in standard position on the coordinate plane:

• (a) $\theta = 135^\circ$

• (b) $\theta = -\frac{5\pi}{6}$

Background

Topic: Angles in Standard Position

This question tests your understanding of how to represent angles in standard position on the coordinate plane, including positive and negative angles and their terminal sides.

Key Terms:

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

• Terminal Side: The position of the ray after rotation.

• Positive Angle: Measured counterclockwise from the positive x-axis.

• Negative Angle: Measured clockwise from the positive x-axis.

Step-by-Step Guidance

1. For $\theta = 135^\circ$, start at the positive x-axis and rotate counterclockwise 135 degrees. Determine which quadrant the terminal side will be in.

2. For $\theta = -\frac{5\pi}{6}$, first convert the angle to degrees if needed: $-\frac{5\pi}{6}$ radians $= -150^\circ$. Rotate clockwise 150 degrees from the positive x-axis. Identify the quadrant for the terminal side.

3. On the coordinate plane, draw the initial side along the positive x-axis for both angles, then sketch the terminal side according to your rotation.

4. Label each angle and the quadrant where its terminal side lies.

Try sketching these angles on your own before checking your work!

Q2. The terminal side of an angle $\theta$ passes through the point $(-5, 12)$. Find the exact values of $\sin \theta$, $\cos \theta$, and $\tan \theta$. Rationalize denominators if needed.

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to use the coordinates of a point on the terminal side of an angle to find the sine, cosine, and tangent values.

Key Terms and Formulas:

• Reference Triangle: A right triangle formed by dropping a perpendicular from the point to the x-axis.

• Trigonometric Ratios:

$\sin \theta = \frac{y}{r}$ $\cos \theta = \frac{x}{r}$ $\tan \theta = \frac{y}{x}$ Where $r = \sqrt{x^2 + y^2}$

Step-by-Step Guidance

1. Identify $x = -5$ and $y = 12$ from the given point.

2. Calculate $r$ using the Pythagorean Theorem: $r = \sqrt{(-5)^2 + 12^2}$.

3. Write the expressions for $\sin \theta$, $\cos \theta$, and $\tan \theta$ using the formulas above.

4. Simplify each ratio, and if necessary, rationalize any denominators.

Try calculating the values before checking the answer!

Q3. If $\cos \theta = -\frac{3}{5}$ and $\theta$ is in Quadrant III, find $\sin \theta$ and $\tan \theta$. Then verify the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$.

Background

Topic: Trigonometric Functions and Identities

This question tests your ability to find other trigonometric values given one value and the quadrant, and to verify a fundamental trigonometric identity.

Key Terms and Formulas:

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

• Sign of Functions in Quadrants: In Quadrant III, both sine and cosine are negative, tangent is positive.

• Relationship: $\sin \theta = \pm \sqrt{1 - \cos^2 \theta}$

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Step-by-Step Guidance

1. Since $\cos \theta = -\frac{3}{5}$ and $\theta$ is in Quadrant III, determine the sign of $\sin \theta$.

2. Use the Pythagorean identity to solve for $\sin \theta$:

3. $\sin^2 \theta = 1 - \cos^2 \theta$

4. Take the square root, remembering the sign based on the quadrant.

5. Set up $\tan \theta = \frac{\sin \theta}{\cos \theta}$, and prepare to verify the identity by plugging in your values.

Try working through the calculations before checking the answer!

Q4. Given that $\cos x = 0.7$, evaluate: $\cos(x - 2\pi) + 3\cos(x + 4\pi) - \cos(x + 6\pi)$

Background

Topic: Trigonometric Function Periodicity

This question tests your understanding of the periodic properties of the cosine function.

Key Terms and Formulas:

• Periodicity of Cosine: $\cos(x + 2\pi k) = \cos x$ for any integer $k$.

Step-by-Step Guidance

1. Recognize that adding or subtracting multiples of $2\pi$ to the argument of cosine does not change its value.

2. Rewrite each term using the periodicity property:

3. $\cos(x - 2\pi) = \cos x$ $\cos(x + 4\pi) = \cos x$ $\cos(x + 6\pi) = \cos x$

4. Substitute $\cos x = 0.7$ into each term and set up the expression for evaluation.

Try simplifying the expression before checking the answer!

Q5. A circle has radius $r = 5$ cm and a central angle $\theta = 72^\circ$. Find:

• (a) the arc length $s$ cut off by $\theta$

• (b) the area $A$ of the sector

Background

Topic: Arc Length and Sector Area

This question tests your ability to use formulas for arc length and sector area, requiring conversion of degrees to radians.

Key Terms and Formulas:

• Arc Length: $s = r\theta$ (with $\theta$ in radians)

• Sector Area: $A = \frac{1}{2} r^2 \theta$ (with $\theta$ in radians)

• Degree to Radian Conversion: $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$

Step-by-Step Guidance

1. Convert $\theta = 72^\circ$ to radians using $\theta_{\text{rad}} = 72 \times \frac{\pi}{180}$.

2. For arc length, use $s = r\theta$ with $r = 5$ cm and your radian value for $\theta$.

3. For sector area, use $A = \frac{1}{2} r^2 \theta$ with the same values.

4. Set up the expressions for $s$ and $A$ but do not compute the final values yet.

Try setting up the calculations before checking the answer!

Q6. Simplify using cofunction identities: $\frac{\sin 25^\circ}{\cos 65^\circ} + \frac{\cos 40^\circ}{\sin 50^\circ}$

Background

Topic: Cofunction Identities

This question tests your knowledge of cofunction identities for sine and cosine.

Key Terms and Formulas:

• Cofunction Identities:

$\sin(90^\circ - x) = \cos x$ $\cos(90^\circ - x) = \sin x$

Step-by-Step Guidance

1. Recognize that $\cos 65^\circ = \sin(90^\circ - 65^\circ) = \sin 25^\circ$ and $\sin 50^\circ = \cos(40^\circ)$.

2. Rewrite the expression using these identities.

3. Simplify each fraction and then add the results.

Try simplifying the expression before checking the answer!

Q7. Evaluate exactly: $\sin(-30^\circ) + \cos(-60^\circ) + \tan(-45^\circ)$

Background

Topic: Trigonometric Values of Negative Angles

This question tests your understanding of the even-odd properties of trigonometric functions and your ability to evaluate them at standard angles.

Key Terms and Formulas:

• Even-Odd Properties:

$\sin(-x) = -\sin x$ $\cos(-x) = \cos x$ $\tan(-x) = -\tan x$

Step-by-Step Guidance

1. Apply the even-odd properties to rewrite each term in terms of positive angles.

2. Recall the exact values for $\sin 30^\circ$, $\cos 60^\circ$, and $\tan 45^\circ$.

3. Substitute these values into the expression and set up the sum.

Try evaluating each term before checking the answer!

Q8. For the function $f(x) = -4 \sin\left(\frac{1}{3}x\right)$:

• (a) Find the amplitude and period.

• (b) Sketch one full cycle of the graph.

Background

Topic: Graphs of Sine Functions

This question tests your understanding of how to determine amplitude and period for sine functions, especially with transformations.

Key Terms and Formulas:

• Amplitude: $|A|$ where $A$ is the coefficient of $\sin$.

• Period: $\frac{2\pi}{B}$ where $B$ is the coefficient of $x$ inside the sine function.

Step-by-Step Guidance

1. Identify $A = -4$ and $B = \frac{1}{3}$ from the function.

2. Calculate the amplitude as $|A|$.

3. Calculate the period as $\frac{2\pi}{B}$.

4. Set up the x-values for one full cycle and describe the key points (max, min, intercepts) for the graph.

Try finding the amplitude and period before sketching the graph!

Q9. For $g(x) = 3 \cos(-2x)$:

• (a) Find amplitude and period.

• (b) Sketch two full periods of the graph.

Background

Topic: Graphs of Cosine Functions

This question tests your ability to analyze amplitude and period for cosine functions, including the effect of negative coefficients.

Key Terms and Formulas:

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

Step-by-Step Guidance

1. Identify $A = 3$ and $B = -2$ from the function.

2. Calculate the amplitude as $|A|$.

3. Calculate the period as $\frac{2\pi}{|B|}$.

4. Set up the x-values for two full periods and describe the key points for the graph.

Try determining amplitude and period before sketching the graph!

Q10. Using an isosceles right triangle, derive the exact values of $\sin 45^\circ$, $\cos 45^\circ$, and $\tan 45^\circ$. Show all steps.

Background

Topic: Special Right Triangles and Exact Trig Values

This question tests your understanding of how to use geometric reasoning to derive exact trigonometric values for $45^\circ$.

Key Terms and Formulas:

• Isosceles Right Triangle: A triangle with two equal sides and angles of $45^\circ$, $45^\circ$, and $90^\circ$.

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Trig Ratios: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Step-by-Step Guidance

1. Draw an isosceles right triangle with legs of length 1.

2. Use the Pythagorean Theorem to find the hypotenuse: $c = \sqrt{1^2 + 1^2}$.

3. Write the expressions for $\sin 45^\circ$, $\cos 45^\circ$, and $\tan 45^\circ$ using the side lengths.

4. Simplify the ratios to their exact values.

Try deriving the values before checking the answer!

Q11. Find the reference angle for each, then evaluate the trig function exactly:

• (a) $\sin 210^\circ$

• (b) $\cos \frac{5\pi}{4}$

• (c) $\tan(-120^\circ)$

Background

Topic: Reference Angles and Exact Trig Values

This question tests your ability to find reference angles and use them to evaluate trigonometric functions exactly.

Key Terms and Formulas:

• Reference Angle: The acute angle formed by the terminal side of the given angle and the x-axis.

• Signs in Quadrants: Know which trig functions are positive/negative in each quadrant.

Step-by-Step Guidance

1. For each angle, determine its quadrant and find the reference angle.

2. Recall the exact value for the trig function at the reference angle.

3. Apply the correct sign based on the quadrant.

4. Set up the expression for the exact value.

Try finding the reference angles and values before checking the answer!

Q12. Compute the exact value: $\frac{\cot 30^\circ + \sin 30^\circ}{\cos 60^\circ - \tan 45^\circ}$

Background

Topic: Exact Trigonometric Values and Simplification

This question tests your ability to recall exact values for standard angles and simplify expressions involving them.

Key Terms and Formulas:

• $\cot 30^\circ = \frac{1}{\tan 30^\circ}$

• Recall $\sin 30^\circ$, $\cos 60^\circ$, $\tan 45^\circ$

Step-by-Step Guidance

1. Write the exact values for $\cot 30^\circ$, $\sin 30^\circ$, $\cos 60^\circ$, and $\tan 45^\circ$.

2. Substitute these values into the expression.

3. Simplify the numerator and denominator separately.

4. Set up the final simplified fraction.

Try simplifying the expression before checking the answer!

Q13. Simplify the expression: $\tan 25^\circ \cdot \cot 25^\circ + \sin^2 40^\circ + \cos^2 40^\circ + \sin\left(\frac{\pi}{2}\right)$

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to use basic trigonometric identities to simplify expressions.

Key Terms and Formulas:

• $\tan x \cdot \cot x = 1$

• $\sin^2 x + \cos^2 x = 1$

• $\sin\left(\frac{\pi}{2}\right) = 1$

Step-by-Step Guidance

1. Recognize that $\tan 25^\circ \cdot \cot 25^\circ = 1$.

2. Recall that $\sin^2 40^\circ + \cos^2 40^\circ = 1$ by the Pythagorean identity.

3. Recall that $\sin\left(\frac{\pi}{2}\right) = 1$.

4. Add the results together to set up the final sum.

Try simplifying the expression before checking the answer!