Exam 6: Analytic Trigonometry

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\cos \theta = - \frac { 3 } { 5 } , \quad \sin \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$ .

Find the exact value of the expression. Do not use a calculator. - $\tan ^ { - 1 } \left[ \tan \left( \frac { 6 \pi } { 7 } \right) \right]$

Solve the problem. -The formula $\mathrm { D } = 24 \left[ 1 - \frac { \cos ^ { - 1 } ( \tan \mathrm { i } \tan \theta ) } { \pi } \right]$ can be used to approximate the number of hours of daylight when the declination of the sun is $i ^ { \circ }$ at a location $\theta ^ { \circ }$ latitude for any date between the vernal equinox and autumnal equinox. To use this formula, $\cos ^ { - 1 }$ (tan i tan $\theta$ ) must be expressed in radians. Approximate the number of hours of daylight in Fargo, North Dakota, (4652'north latitude) for vernal equinox $\left( \mathrm { i } = 0 ^ { \circ } \right)$ . orth

Find the exact value of the expression. - $\sin ^ { - 1 } ( 0.5 )$

Find the exact value under the given conditions. - $\sin \alpha = \frac { 3 } { 5 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \cos \beta = \frac { 2 } { 5 } , 0 < \beta < \frac { \pi } { 2 }$ Find $\cos ( \alpha - \beta )$

Find the exact value of the expression. Do not use a calculator. - $\cos ^ { - 1 } \left[ \cos \left( \frac { \pi } { 10 } \right) \right]$

Solve the problem. -When light travels from one medium to another-from air to water, for instance-it changes direction. (This is why a pencil, partially submerged in water, looks as though it is bent.) The angle of incidence $\theta _ { \mathrm { i } }$ is the angle in the first medium; the angle of refraction $\theta _ { \mathrm { r } }$ is the second medium. (See illustration.) Each medium has an index of refraction $- n _ { i }$ and $n _ { r }$ , respectively-which can be found in tables. Snell's law relates these quantities in the forr $\mathrm { n } _ { \mathrm { i } } \sin \theta _ { \mathrm { i } } = \mathrm { n } _ { \mathrm { r } } \sin \theta _ { \mathrm { r } }$ Solving for $\theta _ { r }$ , we obtain $\theta _ { \mathrm { r } } = \sin ^ { - 1 } \left( \frac { \mathrm { n } _ { \mathrm { i } } } { \mathrm { n } _ { \mathrm { r } } } \sin \theta _ { \mathrm { i } } \right)$ Find $\theta _ { \mathrm { r } }$ for crown glass $\left( \mathrm { n } _ { \mathrm { i } } = 1.52 \right)$ , water( $\left. \mathrm { n } _ { \mathrm { r } } = 1.33 \right)$ , and $\theta _ { \mathrm { i } } = 38 ^ { \circ }$ .

Find the exact value of the expression. - $\sec ^ { - 1 } ( \sqrt { 2 } )$

Solve the problem. -Rewrite over a common denominator: $\frac { 1 } { 1 - \sin \theta } + \frac { 1 } { 1 + \sin \theta }$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $6 \csc \theta - 2 = 4$

Solve the problem. -A weight is suspended on a system of spring and oscillates up and down according to $P = 0.1 [ 3 \cos ( 8 t ) - \sin ( 8 t ) ]$ where P is the position in meters above or below the point of equilibrium (P = 0) and t is time in seconds. Find the time when the weight is at equilibrium. Find all values of $0 \leq t \leq 1$ 1, rounded to the nearest 0.01 second.

Find the exact value of the expression. - $\sec \left[ \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right]$

Find the exact value of the expression.884:890 - $\cos \left[ \sin ^ { - 1 } \frac { 2 } { 3 } + 2 \sin ^ { - 1 } \left( - \frac { 1 } { 3 } \right) \right]$

Find the exact value of the expression. - $\tan ^ { - 1 } ( 1 )$

Express the product as a sum containing only sines or cosines. - $\sin ( 5 \theta ) \sin ( 2 \theta )$

Complete the identity. - $2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 } = ?$

Solve the equation on the interval $0 \leq \theta < 2 \pi \text {. }$ - $\sqrt { 3 } \sin \theta - \cos \theta = - 1$

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places. - $\sin \theta = 0,29$

Find the exact value under the given conditions. - $\sin \alpha = \frac { 7 } { 25 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \cos \beta = \frac { 20 } { 29 } , 0 < \beta < \frac { \pi } { 2 } \quad$ Find $\cos ( \alpha + \beta )$ .

Write the trigonometric expression as an algebraic expression in u. - $\sin \left( \csc ^ { - 1 } \mathrm { u } \right)$