6. Trigonometric Identities and More Equations

Find the exact value of the expression.

$\cos105\degree$

Find the exact value of the expression.

$\sin15\degree$

Find the exact value of the expression.

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

Find the exact value of the expression.

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

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

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

Find the exact value of the expression.

$\tan105\degree$

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

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

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

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

cos(-15°)

Find the exact value of each expression.

sin 255°

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

cos 105° (Hint: 105° = 60° + 45°)

Find the exact value of each expression.

tan 285°

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

cos π/12

Find the exact value of each expression.

sin (13π/12)

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

cos (-7π/12)

Find the exact value of each expression.

tan (5π/12)

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

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

Find the exact value of each expression.

sin (π/12)

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

sin 15°

Find the exact value of each expression.

sin (- 5π/12)

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

sin (2π/5)

Find the exact value of each expression.

tan (-7π/12)

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

sin 142° 14'

Find the exact value of each expression.

sin (-13π/12)

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

cot (9π/10)

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

sin 40° cos 50° + cos 40° sin 50°

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

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

tan 174° 03'

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

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

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

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

sin 98.0142°

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

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

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

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

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

sin 2π/3 = _____ (- π/6)

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

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

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

____ 72° = cot 18°

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

cos(θ - 30°)

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

tan 24° = 1/ _____ 66°

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

cos(45° - θ)

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

tan θ = cot(45° + 2θ)

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

sin θ = cos(2θ + 30°)

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

sin(45° + θ)

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

sec x = csc (2π/3)

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

cos x = sin (π/12)

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

tan (π/4 + x)

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

sin (3π/4 - x)

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)

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

tan(180° + θ)

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°)

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

sin(π + x)

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

cos 3x

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)

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

cos 4x

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

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

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​​

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

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

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

sin 60° cos 45° - cos 60° sin 45°

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

cos( π/2 + x) = -sin x

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

cos 2x = cos² x - sin² x

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

cos 2x = 1 - 2 sin² x

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

cos 2x = (cot² x - 1)/(cot² x + 1)

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))

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

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

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

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

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

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

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

Verify that each equation is an identity.

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

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)

Verify that each equation is an identity.

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

Verify that each equation is an identity.

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

Verify that each equation is an identity.

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

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

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

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

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

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

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

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

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

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

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

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