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6. Trigonometric Identities and More Equations

Sum and Difference Identities

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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How to evaluate the difference of two angles for cosine function

How to evaluate the difference of two angles for cosine function

Cosine of the Difference of Two Angles

Cosine of the Difference of Two Angles

Cosine of Difference

Cosine of Difference

Professor Cornell

Cosine of the Difference of Two Angles

Cosine of the Difference of Two Angles

Using the difference of two angles to evaluate the an angle for cosine, cos195

Using the difference of two angles to evaluate the an angle for cosine, cos195

How to use the Cosine Sum and Difference Identities to Find Exact Values

How to use the Cosine Sum and Difference Identities to Find Exact Values

Math and Stats Help

Using The Sum and Difference Identity to Determine a Sine Function Value

Using The Sum and Difference Identity to Determine a Sine Function Value

Sum and Difference Formulas

Sum and Difference Formulas

Mario's Math Tutoring

Using Tangent Sum and Difference Identities to Find an Exact Value

Using Tangent Sum and Difference Identities to Find an Exact Value

Math and Stats Help

Sum and Difference Identities for Tangent

Sum and Difference Identities for Tangent

Tangent Sum and Difference Formulas How to Use

Tangent Sum and Difference Formulas How to Use

Mario's Math Tutoring

Write sum and difference formula for tangent as one trig function

Write sum and difference formula for tangent as one trig function

Tangent Sum & Difference Identities

Tangent Sum & Difference Identities

Stephanie Yurasits

Showing 13 of 13 videos

Practice this topic

Multiple Choice

Find the exact value of the expression.

$\cos105\degree$

Multiple Choice

Find the exact value of the expression.

$\sin15\degree$

Multiple Choice

Find the exact value of the expression.

$\cos\frac{5\pi}{12}$

Multiple Choice

Find the exact value of the expression.

$\cos80\degree\cos20\degree+\sin80\degree\sin20\degree$

Multiple Choice

Expand the expression using the sum & difference identities and simplify.

$\sin\left(-\theta-\frac{\pi}{2}\right)$

Multiple Choice

Find the exact value of the expression.

$\tan105\degree$

Multiple Choice

Expand the expression using the sum & difference identities and simplify.

$\tan\left(-\theta-\frac{\pi}{2}\right)$

Multiple Choice

Find $\cos\left(a+b\right)$ given $\cos a=\frac12$ , $\sin b=\frac12$ , & $a$ is in Q IV and $b$ is in Q II.

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. sin x cos 2x

Textbook Question

Use 105° = 135° - 30° to find the exact value of 105°.

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos(-15°)

Textbook Question

Find the exact value of each expression.

sin 255°

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos 105° (Hint: 105° = 60° + 45°)

Textbook Question

Find the exact value of each expression.

tan 285°

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos π/12

Textbook Question

Find the exact value of each expression.

sin (13π/12)

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos (-7π/12)

Textbook Question

Find the exact value of each expression.

tan (5π/12)

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)

cos (7π/9) cos (2π/9) - sin (7π/9) sin (2π/9)

Textbook Question

Find the exact value of each expression.

sin (π/12)

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

sin 15°

Textbook Question

Find the exact value of each expression.

sin (- 5π/12)

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

sin (2π/5)

Textbook Question

Find the exact value of each expression.

tan (-7π/12)

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

sin 142° 14'

Textbook Question

Find the exact value of each expression.

sin (-13π/12)

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

cot (9π/10)

Textbook Question

Find the exact value of each expression. See Example 1.

sin 40° cos 50° + cos 40° sin 50°

Textbook Question

Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.

sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

tan 174° 03'

Textbook Question

Find the exact value of each expression. See Example 1.

sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .

Textbook Question

Use the given information to find sin(x + y).

sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.

sin 98.0142°

Textbook Question

Find the exact value of each expression. See Example 1.

[tan 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]

Textbook Question

Use the given information to find sin(x + y).

cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III

Textbook Question

Use identities to fill in each blank with the appropriate trigonometric function name.

sin 2π/3 = _____ (- π/6)

Textbook Question

Find the exact value of each expression. See Example 1.

[tan 5π/12 + tan π/4]/[1 - tan 5π/12 tan π/4]

Textbook Question

Use identities to fill in each blank with the appropriate trigonometric function name.

____ 72° = cot 18°

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

cos(θ - 30°)

Textbook Question

Use identities to fill in each blank with the appropriate trigonometric function name.

tan 24° = 1/ _____ 66°

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

cos(45° - θ)

Textbook Question

Find one value of θ or x that satisfies each of the following.

tan θ = cot(45° + 2θ)

Textbook Question

Find one value of θ or x that satisfies each of the following.

sin θ = cos(2θ + 30°)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin(45° + θ)

Textbook Question

Find one value of θ or x that satisfies each of the following.

sec x = csc (2π/3)

Textbook Question

Find one value of θ or x that satisfies each of the following.

cos x = sin (π/12)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

tan (π/4 + x)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin (3π/4 - x)

Textbook Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(90° - θ)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

tan(180° + θ)

Textbook Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(θ - 270°)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin(π + x)

Textbook Question

Express each function as a trigonometric function of x. See Example 5.

cos 3x

Textbook Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(270° + θ)

Textbook Question

Express each function as a trigonometric function of x. See Example 5.

cos 4x

Textbook Question

Find cos(s + t) and cos(s - t).

cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

Textbook Question

Find cos(s + t) and cos(s - t).

sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV

Textbook Question

Find cos(s + t) and cos(s - t).

cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV

Textbook Question

Match each expression in Column I with its equivalent expression in Column II.

sin 60° cos 45° - cos 60° sin 45°

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos( π/2 + x) = -sin x

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = cos² x - sin² x

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = 1 - 2 sin² x

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = (cot² x - 1)/(cot² x + 1)

Textbook Question

Match each expression in Column I with its equivalent expression in Column II.

(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. 3x x cos -------- sin ------- 2 2

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

Textbook Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°

Textbook Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. 5𝝅 tan --------- 12

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. ( 𝝅 𝝅 ) tan ( ------ + ------ ) ( 3 4 )

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. ( 5𝝅 𝝅 ) tan ( ------ ﹣ ------) ( 3 4 )

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. 5𝝅 𝝅 5𝝅 𝝅 sin ------- cos -------- ﹣ cos -------- sin ------- 12 4 12 4

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 3 5 sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II. 5 13

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 3 5 sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II. 5 13

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 8 1 cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III. 17 2

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 8 1 cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III. 17 2

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- . 6 2 7 2

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- . 6 2 7 2

Textbook Question

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β

Textbook Question

Use the given information to find cos(x - y).

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

Textbook Question

Use the given information to find tan(x + y).

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

Textbook Question

Use the given information to find the quadrant of x + y.

sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in quadrant III

Textbook Question

Use the given information to cos(x - y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Textbook Question

Use the given information to find tan(x + y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Textbook Question

Use the given information to find the quadrant of x + y.

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Textbook Question

Verify that each equation is an identity.

sin(x + y) + sin(x - y) = 2 sin x cos y

Textbook Question

Verify that each equation is an identity. See Example 4.

tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

Textbook Question

Verify that each equation is an identity.

sin(s + t)/cos s cot t = tan s + tan t

Textbook Question

Verify that each equation is an identity.

sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)

Textbook Question

Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

Textbook Question

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

x + y = 9, 2x + y = -1

Textbook Question

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

5x - 2y + 4 = 0, 3x + 5y = 6

Textbook Question

Use the given information to find sin(s + t). See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Textbook Question

Use the given information to find tan(s + t). See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Textbook Question

Use the given information to find the quadrant of s + t. See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Textbook Question

Use the given information to find sin(s + t). See Example 3.

cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Textbook Question

Use the given information to find tan(s + t). See Example 3.

cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Textbook Question

Use the given information to find the quadrant of s + t. See Example 3.

cos s = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Textbook Question

Use the given information to find sin(s + t). See Example 3.

cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Textbook Question

Use the given information to find tan(s + t). See Example 3.

cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Textbook Question

Use the given information to find the quadrant of s + t. See Example 3.

cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Textbook Question

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. cos(45° - 30°)

Textbook Question

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. ( 2π π ) cos ------- ﹣ ------ ( 3 6 )

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. cos 50° cos 20° + sin 50° sin 20°

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. cos 50° cos 20° + sin 50° sin 20°

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. 5π π 5π π cos ------- cos -------- + sin -------- sin ------- 12 12 12 12

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. 5π π 5π π cos ------- cos -------- + sin -------- sin ------- 12 12 12 12

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) 3 5 sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II. 5 13

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) 3 1 tan α = ﹣ ------ , α lies in quadrant II, and cos β = ------- , β lies in quadrant I. 4 3

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) 8 1 cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III. 17 2

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) 3 3𝝅 1 3𝝅 tan α = ------ , 𝝅 < α < -------- , and cos β = ------- , ---------- < β < 2𝝅. 4 2 4 2

Textbook Question

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. sin 6x sin 2x

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. cos 7x cos 3x

Textbook Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: β e. cos ------- 2 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions: β e. cos ------- 2 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2

Textbook Question

Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- < α < 𝝅 5 2 5 𝝅 cos β = ------ , 0 < β < ------ 13 2 Find the exact value of each of the following. cos (α + β)

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