6. Trigonometric Identities and More Equations

Find all solutions to the equation where 0 ≤ $\(\theta\)$ ≤ $2\(\pi\)$.

$\(\sin\]\theta\[\cos\]\left\)(2\(\theta\[\right\))-\(\sin\]\left\)(2\(\theta\[\right\))\(\cos\]\theta\)=\(\frac{\sqrt2}{2}\)$

In Exercises 35–38, find the exact value of the following under the given conditions:

e. cos( β/2)

sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.

sin² θ + 3 sin θ + 2 = 0

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x + cos x = 1

In Exercises 55–58, use the given information to find the exact value of each of the following:

b. cos(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. sin 105°

In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x

Find all solutions to the equation.

$\(\left\)(\(\cos\[\theta\)+\(\sin\]\theta\[\right\))\(\left\)(\(\cos\]\theta\)-\(\sin\[\theta\]\right\))=-\(\frac\)12$

Solve each equation for all exact solutions, in degrees.

tan θ - sec θ = 1

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

5 + 5 tan² θ = 6 sec θ

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

6 sin² θ + sin θ = 1

In Exercises 59–68, verify each identity.

cos²(θ/2) = (sec θ + 1)/(2 sec θ)

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.

3 cos² θ + 2 cos θ - 1 = 0

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 cos² x + cos x ― 1 = 0