Exam 5: Analytic Trigonometry

Solve the problem. -Two factories blow their whistles at exactly the same time. If a man hears the two blasts exactly $6.3$ seconds and $5.2$ seconds after they are blown and the angle between his lines of sight to the two factories is $62.6 ^ { \circ }$ , how far apart are the factories? Give your result to the nearest meter. (Use the fact that sound travels at $344 \mathrm {~m} / \mathrm { sec }$ )

A trigonometry student makes the statement, "If we know any two sides and one angle of a triangle, then the triangle is uniquely determined." Is this a valid statement? Explain, referring to the congruence axioms.

Find the exact value by using a half-angle identity. - $\sin 22.5 ^ { \circ }$

Solve the triangle. - $\mathrm { C } = 101 ^ { \circ } , \mathrm { b } = 43 , \mathrm { c } = 63$

Find the exact value by using a half-angle identity. - $\cos \left( - \frac { \pi } { 8 } \right)$

Simplify the expression to either 1 or -1. - $\tan ^ { 2 } x - \sec ^ { 2 } ( - x )$

Write the expression as the sine, cosine, or tangent of an angle. - $\sin 4 x \cos x - \cos 4 x \sin x$

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 21.3 b = 14.7 C = 14.9

Provide an appropriate response. -A student claims: "Since $1 + \cot ^ { 2 } \theta = \csc ^ { 2 } \theta$ , I should be able to also say $1 + \cot \theta = \csc \theta$ if I take the square root of both sides." Comment on the student's statement.

Prove the identity. - $\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)$

Prove the identity. - $\cos ^ { 2 } x - \sin ^ { 2 } x = 1 - 2 \sin ^ { 2 } x$

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. - $\mathrm { C } = 40 ^ { \circ } , \mathrm { a } = 19 , \mathrm { c } = 10$

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $1 - \sin ^ { 3 } x$