Trigonometry Exam Instructions
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Questions and Answers

What is the simplified form of the equation cos(x) cos(2x) + sin(x) sin(2x) over the interval [0, 2π)?

  • tan(3x)
  • sin(3x)
  • cos(3x) (correct)
  • sec(3x)

Which side of the right triangle is conventionally labeled as 'x' in relation to angle q?

  • Hypotenuse
  • Opposite side
  • Base
  • Adjacent side (correct)

Based on the Pythagorean Theorem x² + y² = r², which of the following equations accurately relates sec q to tan q?

  • sec² q = tan q + 1
  • sec² q = tan² q - 1
  • sec² q = 1 + tan² q (correct)
  • sec q = 1/tan q

What value does r represent in the context of the Pythagorean Theorem?

<p>The length of the hypotenuse (C)</p> Signup and view all the answers

Which trigonometric identity can be used to transform cos(x) cos(2x) + sin(x) sin(2x) into another form?

<p>Sum-to-Product Identity (D)</p> Signup and view all the answers

What is required to receive full credit for answers on this exam?

<p>Clear explanation and supporting work (A)</p> Signup and view all the answers

Which of the following sections does NOT have Eugenia Malitsky as the instructor?

<p>Section 003 (D)</p> Signup and view all the answers

What type of problems does the exam consist of?

<p>Both multiple-choice and written problems (A)</p> Signup and view all the answers

Which statement about the use of materials during the exam is incorrect?

<p>Textbooks may be used. (D)</p> Signup and view all the answers

What could result in receiving little or no credit according to the exam guidelines?

<p>Submitting unsupported answers (D)</p> Signup and view all the answers

What does partial credit depend on for incorrect solutions?

<p>The presentation of work (C)</p> Signup and view all the answers

What is the maximum total score obtainable on the written problems section of the exam?

<p>80 points (C)</p> Signup and view all the answers

What is emphasized about the organization of work in the exam?

<p>It should be neat and coherent. (A)</p> Signup and view all the answers

Which expression is equivalent to $1 - 2 ext{sin}^2 x + ext{sin}^4 x$?

<p>$ ext{sin}^2 x$ (B)</p> Signup and view all the answers

What is the range of solutions for the equation $2 ext{sin}^2 x - 1 = 0$?

<p>[0, 4\pi) (D)</p> Signup and view all the answers

For which of the following values of $q$ does the equation $\text{csc} q = 2$ hold true?

<p>$\frac{\pi}{6}$ (D)</p> Signup and view all the answers

Which of the following is the solution set for $2 ext{sin}^2 x - 1 = 0$ in the interval [0, 2π)?

<p>${ \frac{\pi}{6}, \frac{5\pi}{6} }$ (B)</p> Signup and view all the answers

What transformation of the sine function is represented by the equation $\text{csc} q = 2$?

<p>Vertical stretch (D)</p> Signup and view all the answers

Which angle is not a solution to the equation $\text{csc} q = 2$?

<p>$\frac{3\pi}{6}$ (B)</p> Signup and view all the answers

What method will give you the solutions to $\text{sin}^4 x + \text{cos}^2 x = 1$?

<p>Using identities (A)</p> Signup and view all the answers

Which function is used to find the solutions of the equation $2 ext{sin}^2 x - 1 = 0$?

<p>$\text{sin}$ (C)</p> Signup and view all the answers

Which transformation correctly describes how g(x) is derived from f(x) given by g(x) = cos(2x p) + 3?

<p>g(x) is obtained by shifting f(x) upward 3 units. (D)</p> Signup and view all the answers

In which interval does the graph of g(x) contain exactly one period?

<p>[p, 2p] (C)</p> Signup and view all the answers

What is the relationship between the periods of f(x) and g(x)?

<p>g(x) has a shorter period than f(x). (D)</p> Signup and view all the answers

If g(x) is transformed by shifting f(x) downward p2 units, what will the new function represent?

<p>It will represent a cosine function moved vertically downwards. (C)</p> Signup and view all the answers

What is the equivalent expression for sec(arcsin(2x))?

<p>1/sqrt(1 - (2x)^2) (D)</p> Signup and view all the answers

Which transformation would NOT correctly derive g(x) from f(x)?

<p>Multiplying f(x) by 2. (A)</p> Signup and view all the answers

What does the transformation cos(2x p) imply about the speed of oscillation of g(x)?

<p>It oscillates at a faster rate than f(x). (C)</p> Signup and view all the answers

What does the term 'exactly one period' entail for the function g(x)?

<p>g(x) completes one full cycle. (D)</p> Signup and view all the answers

What is the fundamental identity representing the relationship between sine and cosine?

<p>cos²(q) + sin²(q) = 1 (D)</p> Signup and view all the answers

In a triangle where AB = 8, BC = 7, and AC = 5, which formula is used to find the angle A?

<p>cos A = (AB² + AC² - BC²) / (2 * AB * AC) (A)</p> Signup and view all the answers

Which of the following is NOT a result of the double angle formulas?

<p>sin(2a) = sin2(a) + cos2(a) (A)</p> Signup and view all the answers

What does the equation tan²(q) + 1 equal?

<p>sec²(q) (A)</p> Signup and view all the answers

If cos²(a) + sin²(a) = 1, which of the following conclusions can be drawn?

<p>Sine squared can never equal 1 when cosine squared equals 0. (A)</p> Signup and view all the answers

Using the law of sines, which quantity can be calculated if the lengths of all three sides of a triangle are known?

<p>Any angle of the triangle (A)</p> Signup and view all the answers

What is the transformed version of cos(2a) according to the identity?

<p>cos(2a) = 2cos²(a) - 1 (B)</p> Signup and view all the answers

Which of the following represents the law of cosines?

<p>c² = a² + b² - 2ab cos(C) (D)</p> Signup and view all the answers

What is the radian measure of the central angle for a circle of radius 3 inches and arc length 21 inches?

<p>$7\pi$ rad (A)</p> Signup and view all the answers

What is the value of $\cos(\frac{11\pi}{6})$?

<p>$\frac{1}{2}$ (A)</p> Signup and view all the answers

If $\cos(t) = \frac{5}{6}$, what is $\sin(t + \frac{\pi}{2})$?

<p>$\frac{1}{6}$ (A)</p> Signup and view all the answers

What is the angle of elevation of the sun if a 5m tall light post casts a shadow of 5√3m?

<p>$60^\circ$ (A)</p> Signup and view all the answers

What is the exact value of $\cot(q)$ if the terminal side of $q$ lies on the line $3x + 4y = 0$ in the third quadrant?

<p>$-\frac{4}{3}$ (A)</p> Signup and view all the answers

For the circle with radius r = 3 inches and an arc length of 21 inches, which of the following statements is true?

<p>The central angle corresponds to 7π radians. (A)</p> Signup and view all the answers

What is the principal value of $\sin(\frac{2\pi}{3})$?

<p>$\frac{\sqrt{3}}{2}$ (A)</p> Signup and view all the answers

Determine the value of $\sin^2(t) + \cos^2(t)$ based on the given value of $\cos(t) = \frac{5}{6}$.

<p>$1$ (C)</p> Signup and view all the answers

Flashcards

Complementary Angles

Two angles are complementary if their sum is 90 degrees.

Finding the Complement

To find the complement of an angle, subtract the angle from 90 degrees.

Understanding 'p' in angle notation

'p' represents the angle measure, while the number indicates the angle's position in a diagram.

Angle Notation

Angle names often include a lowercase letter (p) followed by a number (4) indicating the angle's position in a diagram.

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Multiple Choice Questions

Questions with multiple answer options, only one of which is correct.

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Written Questions

Questions requiring a detailed written solution, often involving calculations and explanations.

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Showing Work

Demonstrating the process used to arrive at an answer, including calculations and reasoning.

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Partial Credit

Credit awarded for a partially correct answer, even if not entirely right.

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Trigonometric Identity

An equation that is true for all values of the variables involved. It allows for the simplification and manipulation of trigonometric expressions.

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Double Angle Formula

A specific type of trigonometric identity that relates the trigonometric functions of an angle (x) to the trigonometric functions of twice that angle (2x).

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Solving Trigonometric Equations

Finding all possible values of an unknown angle (x) that satisfy a given trigonometric equation. This involves using identities to simplify the equation and isolate the unknown angle.

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Pythagorean Theorem

A fundamental theorem in geometry that states the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It relates the lengths of the sides of any right triangle.

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Secant and Tangent Relationship

The secant of an angle (sec q) is the reciprocal of the cosine of that angle (cos q). The tangent of an angle (tan q) is the ratio of the sine (sin q) to the cosine (cos q) of that angle. Using the Pythagorean Theorem, we can relate these two functions by deriving an equation that connects them.

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Radian Measure of a Central Angle

The radian measure of a central angle of a circle is the ratio of the arc length to the radius of the circle.

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Calculating Radian Measure

To find the radian measure of a central angle, use the formula: θ = s/r where θ is the angle in radians, s is the arc length, and r is the radius of the circle.

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Cosine of 11π/6

The cosine of 11π/6 is the x-coordinate of the point on the unit circle that corresponds to the angle 11π/6. This angle is in the fourth quadrant and its reference angle is π/6.

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Sine of (t + π/2)

Using the trigonometric identities, sin(t + π/2) = cos(t).

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Angle of Elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal.

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Calculating Angle of Elevation

Use the tangent function (tan θ = opposite/adjacent) to calculate the angle of elevation. The opposite side is the height of the object, and the adjacent side is the length of the shadow.

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Cotangent: Terminal Side in Third Quadrant

Cotangent is the reciprocal of tangent, so cot θ = 1/tan θ. To find cot θ when the terminal side of θ lies on a line in the third quadrant, find the slope of the line, which is -x/y = tangent of the angle. Then find the reciprocal of the tangent to get the cotangent.

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Cotangent in Terms of x and y

In terms of x and y coordinates, cot θ = x/y.

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Period of a Trigonometric Function

The horizontal distance over which the graph of a trigonometric function repeats itself. For example, the period of the cosine function, f(x) = cos(x), is 2π. This means the graph repeats every 2π units along the x-axis.

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Horizontal Compression/Expansion

The transformation that changes the period of a trigonometric function. If the coefficient of x inside the function is greater than 1, the graph compresses horizontally. If the coefficient is less than 1, the graph expands horizontally.

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Vertical Shift

The transformation that moves the entire graph of a trigonometric function upwards or downwards. The vertical shift is determined by the constant term added to the function. Adding a positive constant shifts the graph upwards, and adding a negative constant shifts the graph downwards.

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Transformations of f(x) = cos(x)

The function g(x) = cos(2xπ) + 3 is a transformation of the parent function f(x) = cos(x). It involves a horizontal compression by a factor of 2π and a vertical shift upwards by 3 units.

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Secant of an Angle

The secant of an angle is the reciprocal of the cosine of that angle. Mathematically, sec(θ) = 1/cos(θ).

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Inverse Sine

The inverse sine function, denoted by arcsin(x) or sin⁻¹(x), gives the angle whose sine is x. In other words, arcsin(x) = θ if and only if sin(θ) = x.

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Finding an Algebraic Expression

Given a trigonometric expression involving inverse trigonometric functions, the goal is to find an equivalent algebraic expression in terms of the variable x. This can be done using trigonometric identities and the definitions of inverse trigonometric functions.

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Sin2x + Sin4x

This trigonometric expression can be simplified using trigonometric identities, often leading to a more convenient form.

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Trigonometric Functions

Functions that relate an angle of a right triangle to the ratio of the lengths of its sides. Examples include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

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Double-Angle Identity

An identity that connects the trigonometric functions of an angle with the trigonometric functions of twice that angle. Examples include sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x).

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Power Reducing Identity

An identity that allows you to express trigonometric functions raised to even powers in terms of functions with lower powers. Examples include sin^2(x) = (1-cos(2x))/2 and cos^2(x) = (1+cos(2x))/2.

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Cosecant (csc)

The reciprocal of the sine function. It's defined as csc(x) = 1/sin(x).

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Solving for Angles in Radians

Finding the values of angles, expressed in radians, that satisfy a given trigonometric equation. Requires understanding the unit circle and trigonometric function values.

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Verifying Identities

The process of manipulating one side of a trigonometric equation to make it identical to the other side.

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Secant (sec)

The reciprocal of the cosine function. sec(x) = 1/cos(x)

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Law of Cosines

A formula that relates the sides and angles of a triangle. c² = a² + b² - 2ab cos(C)

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Finding Angle A

Using the Law of Cosines to solve for an unknown angle in a triangle.

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Fundamental Trigonometric Identities

Basic equations that relate different trigonometric functions. Examples: cos²x + sin²x = 1, tan²x + 1 = sec²x

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Addition and Subtraction Formulas

Formulas that express trigonometric functions of sums and differences of angles. Examples: sin(a + b) = sin(a)cos(b) + cos(a)sin(b), cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

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Study Notes

Exam Instructions

  • Work through all problems, demonstrating your steps. Lack of work results in no credit.
  • Provide clear and detailed explanations for required explanations.
  • You do not need to simplify expressions.
  • The multiple-choice section is worth 20 points.
  • The written section is worth 80 points.
  • No textbooks, notes, cell phones, or calculators are permitted during the exam.
  • All work must be shown on each problem.
  • Organize your work clearly, neatly, and coherently.
  • Unsupported answers will not receive full credit.
  • Partial credit may be awarded for well-reasoned, but incomplete, solutions.
  • Use extra paper if needed, clearly indicating this on the original exam.

Exam Questions (Multiple Choice and Written)

  • Expect problems related to finding complements of angles, measuring central angles in circles, finding values of trigonometric functions, and solving trigonometric equations within a specified interval.
  • Problems will require the application of trigonometric identities and the Pythagorean theorem.

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Math 113 Fall 2023 Final Exam (PDF)

Description

Prepare for your upcoming trigonometry exam with these detailed instructions. The exam will cover finding complements of angles, measuring central angles, and solving trigonometric equations. Be sure to show all your work for partial credit and remember that no external resources are permitted.

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