Lesson 7 Double Identity Solidify Understanding

Ready

Find the two angles, $0 < θ \leq 2 π$ , that are solutions to the equation. Round your answers to $2$ decimals. (Your calculator should be set in radians.)

You do not have to fill out the diagram. It is there to help you think about the solutions.

1.

$\sin θ = \frac{4}{5}$

2.

$\sin θ = \frac{2}{7}$

3.

$\sin θ = - \frac{1}{10}$

4.

$\sin θ = - \frac{13}{14}$

5.

$\cos θ = \frac{11}{12}$

6.

$\cos θ = \frac{1}{9}$

7.

$\cos θ = - \frac{7}{8}$

8.

$\cos θ = - \frac{2}{5}$

Set

The sum and difference formulas are given in the table.

$\sin (u + v) = (\sin u) (\cos v) + (\cos u) (\sin v)$	$\cos (u + v) = (\cos u) (\cos v) - (\sin u) (\sin v)$
$\sin (u - v) = (\sin u) (\cos v) - (\cos u) (\sin v)$	$\cos (u - v) = (\cos u) (\cos v) + (\sin u) (\sin v)$

The sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles. The challenge is recognizing that an unfamiliar angle might be the sum or difference of an angle you know.

Write the following as the sum or difference of two angles for which you know the trig values.

9.

$75 °$

10.

$255 °$

11.

$\frac{π}{12}$

12.

$- \frac{13 π}{12}$

Find the exact value of the following using the sum and difference formulas.

13.

$\sin 75 °$

14.

$\sin 255 °$

15.

$\cos \frac{π}{12}$

16.

$\sin (- \frac{13 π}{12})$

Find the exact value of the function given that $\sin u = \frac{5}{13}$ and $\cos v = - \frac{3}{5}$ .

(Both $u$ and $v$ are in Q. 2.)

17.

$\sin (u + v)$

18.

$\cos (u + v)$

Find the exact value of the function given that $s i n u = - \frac{7}{25}$ and $\cos v = - \frac{4}{5}$ .

(Both $u$ and $v$ are in Q. 3.)

19.

$\sin (u - v)$

20.

$\cos (u - v)$

Go

Find the arc length. Write your answers with $π$ in it. Then use your calculator to find the approximate length of the arc to $2$ decimal places.

21.

Find the length of an arc given that $r = 10 in.$ and $θ = \frac{π}{4}$ .

22.

Find the arc length given $r = 4 cm$ and $θ = \frac{5 π}{6}$ .

23.

Find the arc length given $r = 72.0 ft$ and $θ = \frac{π}{8}$ .

24.

Find the arc length given $r = 50 yd$ and $θ = \frac{8 π}{5}$ .