Stewart Precalc 6e Section 7.2: Addition and Subtraction Formulas

1 Concepts · Level 1

If we know the values of the sine and cosine of \(x\) and \(y\), we can find the value of \(\sin(x + y)\) by using the ______ Formula for Sine. State the formula: \(\sin(x + y) =\) ______.

2 Concepts · Level 1

If we know the values of the sine and cosine of \(x\) and \(y\), we can find the value of \(\cos(x - y)\) by using the ______ Formula for Cosine. State the formula: \(\cos(x - y) =\) ______.

3 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\sin \dfrac{19 \pi}{12}\).

4 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\cos \dfrac{17 \pi}{12}\).

5 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\tan\left(-\dfrac{\pi}{12}\right)\).

6 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\sin\left(-\dfrac{5 \pi}{12}\right)\).

7 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\cos \dfrac{11 \pi}{12}\).

8 Skills - Exact Value (Addition/Subtraction Formula) · Level 2

Use an Addition or Subtraction Formula to find the exact value of the expression: \(\tan \dfrac{7 \pi}{12}\).

9 Skills - Function of One Number · Level 2

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value: \(\cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ}\).

10 Skills - Function of One Number · Level 2

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value: \(\cos \dfrac{3 \pi}{7} \cos \dfrac{2 \pi}{21} + \sin \dfrac{3 \pi}{7} \sin \dfrac{2 \pi}{21}\).

11 Skills - Function of One Number · Level 2

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value: \(\dfrac{\tan \dfrac{\pi}{18} + \tan \dfrac{\pi}{9}}{1 - \tan \dfrac{\pi}{18} \tan \dfrac{\pi}{9}}\).

12 Skills - Function of One Number · Level 2

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value: \(\dfrac{\tan 73^{\circ} - \tan 13^{\circ}}{1 + \tan 73^{\circ} \tan 13^{\circ}}\).

13 Skills - Function of One Number · Level 2

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value: \(\cos \dfrac{13 \pi}{15} \cos\left(-\dfrac{\pi}{5}\right) - \sin \dfrac{13 \pi}{15} \sin\left(-\dfrac{\pi}{5}\right)\).

14 Skills - Cofunction Identity Proof · Level 2

Prove the cofunction identity using the Addition and Subtraction Formulas: \(\tan\left(\dfrac{\pi}{2} - u\right) = \cot u\).

15 Skills - Cofunction Identity Proof · Level 2

Prove the cofunction identity using the Addition and Subtraction Formulas: \(\cot\left(\dfrac{\pi}{2} - u\right) = \tan u\).

16 Skills - Cofunction Identity Proof · Level 2

Prove the cofunction identity using the Addition and Subtraction Formulas: \(\sec\left(\dfrac{\pi}{2} - u\right) = \csc u\).

17 Skills - Cofunction Identity Proof · Level 2

Prove the cofunction identity using the Addition and Subtraction Formulas: \(\csc\left(\dfrac{\pi}{2} - u\right) = \sec u\).

18 Skills - Identity Proof · Level 2

Prove the identity: \(\sin\left(x - \dfrac{\pi}{2}\right) = -\cos x\).

19 Skills - Identity Proof · Level 2

Prove the identity: \(\cos\left(x - \dfrac{\pi}{2}\right) = \sin x\).

20 Skills - Identity Proof · Level 2

Prove the identity: \(\sin(x - \pi) = -\sin x\).

21 Skills - Identity Proof · Level 2

Prove the identity: \(\cos(x - \pi) = -\cos x\).

22 Skills - Identity Proof · Level 2

Prove the identity: \(\tan(x - \pi) = \tan x\).

23 Skills - Identity Proof · Level 2

Prove the identity: \(\sin\left(\dfrac{\pi}{2} - x\right) = \sin\left(\dfrac{\pi}{2} + x\right)\).

24 Skills - Identity Proof · Level 3

Prove the identity: \(\cos\left(x + \dfrac{\pi}{6}\right) + \sin\left(x - \dfrac{\pi}{3}\right) = 0\).

25 Skills - Identity Proof · Level 3

Prove the identity: \(\tan\left(x - \dfrac{\pi}{4}\right) = \dfrac{\tan x - 1}{\tan x + 1}\).

26 Skills - Identity Proof · Level 3

Prove the identity: \(\sin(x + y) - \sin(x - y) = 2 \cos x \sin y\).

27 Skills - Identity Proof · Level 3

Prove the identity: \(\cos(x + y) + \cos(x - y) = 2 \cos x \cos y\).

28 Skills - Identity Proof · Level 3

Prove the identity: \(\cot(x - y) = \dfrac{\cot x \cot y + 1}{\cot y - \cot x}\).

29 Skills - Identity Proof · Level 3

Prove the identity: \(\cot(x + y) = \dfrac{\cot x \cot y - 1}{\cot x + \cot y}\).

30 Skills - Identity Proof · Level 3

Prove the identity: \(\tan x - \tan y = \dfrac{\sin(x - y)}{\cos x \cos y}\).

31 Skills - Identity Proof · Level 3

Prove the identity: \(1 - \tan x \tan y = \dfrac{\cos(x + y)}{\cos x \cos y}\).

32 Skills - Identity Proof · Level 3

Prove the identity: \(\dfrac{\sin(x + y) - \sin(x - y)}{\cos(x + y) + \cos(x - y)} = \tan y\).

33 Skills - Identity Proof · Level 3

Prove the identity: \(\cos(x + y) \cos(x - y) = \cos^2 x - \sin^2 y\).

34 Skills - Identity Proof · Level 3

Prove the identity: \(\sin(x + y + z) = \sin x \cos y \cos z + \cos x \sin y \cos z + \cos x \cos y \sin z - \sin x \sin y \sin z\).

35 Skills - Identity Proof · Level 3

Prove the identity: \(\tan(x - y) + \tan(y - z) + \tan(z - x) = \tan(x - y) \tan(y - z) \tan(z - x)\).

36 Skills - Inverse Trig (in terms of x, y) · Level 3

Write the given expression in terms of \(x\) and \(y\) only: \(\cos(\sin^{-1} x - \tan^{-1} y)\).

37 Skills - Inverse Trig (in terms of x, y) · Level 3

Write the given expression in terms of \(x\) and \(y\) only: \(\tan(\sin^{-1} x + \cos^{-1} y)\).

38 Skills - Inverse Trig (in terms of x, y) · Level 3

Write the given expression in terms of \(x\) and \(y\) only: \(\sin(\tan^{-1} x - \tan^{-1} y)\).

39 Skills - Inverse Trig (in terms of x, y) · Level 3

Write the given expression in terms of \(x\) and \(y\) only: \(\sin(\sin^{-1} x + \cos^{-1} y)\).

40 Skills - Exact Value (Inverse Trig) · Level 3

Find the exact value of the expression: \(\sin\left(\cos^{-1} \dfrac{1}{2} + \tan^{-1} 1\right)\).

41 Skills - Exact Value (Inverse Trig) · Level 3

Find the exact value of the expression: \(\cos\left(\sin^{-1} \dfrac{\sqrt{3}}{2} + \cot^{-1} \sqrt{3}\right)\).

42 Skills - Exact Value (Inverse Trig) · Level 4

Find the exact value of the expression: \(\tan\left(\sin^{-1} \dfrac{3}{4} - \cos^{-1} \dfrac{1}{3}\right)\).

43 Skills - Exact Value (Inverse Trig) · Level 4

Find the exact value of the expression: \(\sin\left(\cos^{-1} \dfrac{2}{3} - \tan^{-1} \dfrac{1}{2}\right)\).

44 Skills - Evaluate Under Conditions · Level 3

Evaluate the expression under the given conditions: \(\cos(\theta - \phi)\); \(\cos \theta = \dfrac{3}{5}\), \(\theta\) in Quadrant IV, \(\tan \phi = -\sqrt{3}\), \(\phi\) in Quadrant II.

45 Skills - Evaluate Under Conditions · Level 3

Evaluate the expression under the given conditions: \(\sin(\theta - \phi)\); \(\tan \theta = \dfrac{4}{3}\), \(\theta\) in Quadrant III, \(\sin \phi = -\dfrac{\sqrt{10}}{10}\), \(\phi\) in Quadrant IV.

46 Skills - Evaluate Under Conditions · Level 3

Evaluate the expression under the given conditions: \(\sin(\theta + \phi)\); \(\sin \theta = \dfrac{5}{13}\), \(\theta\) in Quadrant I, \(\cos \phi = -\dfrac{2 \sqrt{5}}{5}\), \(\phi\) in Quadrant II.

47 Skills - Evaluate Under Conditions · Level 3

Evaluate the expression under the given conditions: \(\tan(\theta + \phi)\); \(\cos \theta = -\dfrac{1}{3}\), \(\theta\) in Quadrant III, \(\sin \phi = \dfrac{1}{4}\), \(\phi\) in Quadrant II.

48 Skills - In Terms of Sine Only · Level 2

Write the expression in terms of sine only: \(-\sqrt{3} \sin x + \cos x\).

49 Skills - In Terms of Sine Only · Level 2

Write the expression in terms of sine only: \(\sin x + \cos x\).

50 Skills - In Terms of Sine Only · Level 2

Write the expression in terms of sine only: \(5(\sin 2x - \cos 2x)\).

51 Skills - In Terms of Sine Only · Level 2

Write the expression in terms of sine only: \(3 \sin \pi x + 3 \sqrt{3} \cos \pi x\).

52 Skills - Express in Sine and Graph · Level 3

(a) Express the function in terms of sine only. (b) Graph the function: \(g(x) = \cos 2x + \sqrt{3} \sin 2x\).

53 Skills - Express in Sine and Graph · Level 3

(a) Express the function in terms of sine only. (b) Graph the function: \(f(x) = \sin x + \cos x\).

54 Skills - Difference Quotient · Level 4

Let \(g(x) = \cos x\). Show that \(\dfrac{g(x + h) - g(x)}{h} = -\cos x \cdot \left(\dfrac{1 - \cos h}{h}\right) - \sin x \cdot \left(\dfrac{\sin h}{h}\right)\).

55 Skills - Show Identity · Level 4

Show that if \(\beta - \alpha = \dfrac{\pi}{2}\), then \(\sin(x + \alpha) + \cos(x + \beta) = 0\).

56 Skills - Figure Proof · Level 4

Refer to the figure. Show that \(\alpha + \beta = \gamma\), and find \(\tan \gamma\).

57 Skills - Slope and Angle of Line · Level 5

(a) If \(L\) is a line in the plane and \(\theta\) is the angle formed by the line and the \(x\)-axis as shown in the figure, show that the slope \(m\) of the line is given by \(m = \tan \theta\). (b) Let \(L_1\) and \(L_2\) be two nonparallel lines in the plane with slopes \(m_1\) and \(m_2\), respectively. Let \(\psi\) be the acute angle formed by the two lines. Show that \(\tan \psi = \dfrac{m_2 - m_1}{1 + m_1 m_2}\). (c) Find the acute angle formed by the two lines \(y = \dfrac{1}{3} x + 1\) and \(y = -\dfrac{1}{2} x - 3\). (d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other.

58 Skills - Graph and Prove · Level 3

(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true: \(y = \sin^2\left(x + \dfrac{\pi}{4}\right) + \sin^2\left(x - \dfrac{\pi}{4}\right)\).

59 Skills - Graph and Prove · Level 3

(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true: \(y = -\dfrac{1}{2}[\cos(x + \pi) + \cos(x - \pi)]\).

60 Skills - Angle Sum from Figure · Level 4

Find \(\angle A + \angle B + \angle C\) in the figure. Hint: First use an addition formula to find \(\tan(A + B)\).

61 Applications - Adding an Echo · Level 3

Adding an Echo. A digital delay device echoes an input signal by repeating it a fixed length of time after it is received. If such a device receives the pure note \(f_1(t) = 5 \sin t\) and echoes the pure note \(f_2(t) = 5 \cos t\), then the combined sound is \(f(t) = f_1(t) + f_2(t)\). (a) Graph \(y = f(t)\) and observe that the graph has the form of a sine curve \(y = k \sin(t + \phi)\). (b) Find \(k\) and \(\phi\).

62 Applications - Interference · Level 3

Interference. Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by \(f_1(t) = C \sin \omega t\) and \(f_2(t) = C \sin(\omega t + \alpha)\). The two sound waves interfere to produce a single sound modeled by the sum: \(f(t) = C \sin \omega t + C \sin(\omega t + \alpha)\). (a) Use the Addition Formula for Sine to show that \(f\) can be written in the form \(f(t) = A \sin \omega t + B \cos \omega t\), where \(A\) and \(B\) are constants that depend on \(\alpha\). (b) Suppose that \(C = 10\) and \(\alpha = \dfrac{\pi}{3}\). Find constants \(k\) and \(\phi\) so that \(f(t) = k \sin(\omega t + \phi)\).

63 Discovery - Discussion - Writing · Level 4

Addition Formula for Sine. In the text we proved only the Addition and Subtraction Formulas for Cosine. Use these formulas and the cofunction identities \(\sin x = \cos\left(\dfrac{\pi}{2} - x\right)\) and \(\cos x = \sin\left(\dfrac{\pi}{2} - x\right)\) to prove the Addition Formula for Sine. Hint: Use the first cofunction identity to write \(\sin(s + t) = \cos(\dfrac{\pi}{2} - (s + t)) = \cos(\left(\dfrac{\pi}{2} - s\right) - t)\), and use the Subtraction Formula for Cosine.

64 Discovery - Discussion - Writing · Level 4

Addition Formula for Tangent. Use the Addition Formulas for Cosine and Sine to prove the Addition Formula for Tangent. Hint: Use \(\tan(s + t) = \dfrac{\sin(s + t)}{\cos(s + t)}\) and divide the numerator and denominator by \(\cos s \cos t\).

65 Example - Using the Addition and Subtraction Formulas · Level 2

Find the exact value of each expression. *(a)* \(\cos(75^{\circ})\) *(b)* \(\cos\left(\dfrac{\pi}{12}\right)\)

66 Example - Using the Addition Formula for Sine · Level 2

Find the exact value of the expression \(\sin(20^{\circ}) \cos(40^{\circ}) + \cos(20^{\circ}) \sin(40^{\circ})\).

67 Example - Proving a Cofunction Identity · Level 2

Prove the cofunction identity \(\cos\left(\dfrac{\pi}{2} - u\right) = \sin u\).

68 Example - Proving an Identity · Level 3

Verify the identity \(\dfrac{1 + \tan x}{1 - \tan x} = \tan\left(\dfrac{\pi}{4} + x\right)\).

69 Example - An Identity from Calculus · Level 3

If \(f(x) = \sin x\), show that \(\dfrac{f(x+h) - f(x)}{h} = \sin x \cdot \dfrac{\cos h - 1}{h} + \cos x \cdot \dfrac{\sin h}{h}\).

70 Example - Simplifying an Expression Involving Inverse Trigonometric Functions · Level 4

Write \(\sin(\cos^{-1} x + \tan^{-1} y)\) as an algebraic expression in \(x\) and \(y\), where \(-1 \leq x \leq 1\) and \(y\) is any real number.

71 Example - Evaluating an Expression Involving Trigonometric Functions · Level 3

Evaluate \(\sin(\theta + \phi)\), where \(\sin \theta = \dfrac{12}{13}\) with \(\theta\) in Quadrant II and \(\tan \phi = \dfrac{3}{4}\) with \(\phi\) in Quadrant III.

72 Example - Sum of Sine and Cosine Terms · Level 3

Express \(3 \sin x + 4 \cos x\) in the form \(k \sin(x + \phi)\).

73 Example - Graphing a Trigonometric Function · Level 3

Write the function \(f(x) = -\sin 2x + \sqrt{3} \cos 2x\) in the form \(k \sin(2x + \phi)\), and use the new form to graph the function.

Quick Answer Input

Submit your exam?