Trigonometry

\(\text{In the triangle shown, what is the value of }\cos x\text{°} ?\)

\(\text{In the right triangle shown, what is the value of }\sin {A}?\)

\(\text{In the right triangle shown, the value of }\tan x\text{°} \text{ is }\frac{c}{39}.\text{ What is the value of }c?\)

\(\begin{aligned}
&\text{In the right triangle shown, }c\text{ is a constant. The value of }\sin x\text{°}=\frac{k}{c}\text{, where }k\\
&\text{is a constant. What of the value of }k?\end{aligned}\)

\(\text{Right triangle }ABC\text{ is shown. What is the value of}\tan A?\)

\(\text{What is the value of }\tan\frac{92\pi}{3}?\)

\(\text{The measure of angle }C\text{ is }\frac{5\pi}{4}\text{ radians. What is the value of }\cos (C)?\)

\(\text{In the triangle shown, }RS=\sqrt{105}.\text{ What is the value of }\sin R?\)

\(\text{Right triangle }ABC\text{ is shown.}\text{ What is the value of }\sin A?\)

\(\text{Triangle }PQR\text{ has right angle }Q.\text{ If }\sin R=\frac{4}{5},\text{ what is the value of }\tan P ?\)

\(\text{In triangle }JKL, \cos(K)=\frac{24}{51}\text{ and angle }J\text{ is a right angle. What is the value of }\cos(L)?\)

\(\text{In the right triangle }JKL\text{, the tangent of }\angle L \text{ is }\frac{3}{4}.\text{ What is the length of }JK?\)

\(\begin{aligned}
&\text{According to a law, ramps for use by the general public must form an angle with level ground }\\
&\text{such that the tangent of the angle }\theta\text{ is }\frac{1}{12}.\text{ In the figure above, if the ramp conforms to the }\\
&\text{law and has a height of 1.5 feet, what is the value of }x\text{, in feet?}\end{aligned}\)

\(\begin{aligned}
&\text{In right triangle }ABC\text{, angles }A\text{ and }B\text{ are acute, side }AC\text{ has a length of }21.7,\\
&\text{and }\tan B =\frac{1}{8}.\text{ What is the length of side }BC \text{, rounded to the nearest tenth?}\end{aligned}\)

\(\begin{aligned}
&\text{In }\triangle ABC,\angle B\text{ is a right angle and the length of }\overline{BC}\text{ is }204\text{ millimeters. If}\\
&\cos A={\frac{3}{5}},\text{what is the length, in millimeters, of }\overline{AB}?\end{aligned}\)

\(\begin{aligned}
&\text{In triangle }XYZ,\text{ angle }Z\text{ is a right angle and the length of }\overline{YZ}\text{ is 24 units.}\\
&\text{If }\tan X=\frac{12}{35}\text{, what is the perimeter, in units, of triangle }XYZ ?\end{aligned}\)

\(\begin{aligned}
&\text{In triangle }XYZ,\text{angle }Y\text{ is a right angle, the measure of angle }Z\text{ is }39^{\circ},\\
&\text{and the length of }\overline{YZ}\text{ is }22\text{ units.}\text{ If the area, in square units, of triangle}\\
&XYZ\text{ can be represented by the expression }k\tan39^{\circ},\text{ where }k\text{ is a }\\&\text{constant, what is the value of }k?\end{aligned}\)

\(\begin{aligned}
&\text{In the figure shown, the measure of angle }X\text{ is }52^{\circ}\text{. The length of }\overline{XY}\text{ is 24 units and the}\\
&\text{length of }\overline{XZ}\text{ is }17\text{ units. What is the area, in square units, of triangle }XYZ?
\end{aligned}\)

\(\begin{aligned}
&\text{In right triangle }RST\text{, the sum of the measures of angle }R\text{ and angle }S\text{ is 90 degrees.}\\
&\text{The value of }\sin(R)\text{ is }\frac{\sqrt{15}}{4}.\text{ What is the value of }\cos(S)? \end{aligned}\)

\(\begin{aligned}
&\text{In a right triangle, one angle measures }x^{\circ},\mathrm{where }\sin x^{\circ}=\frac{4}{5}.\text{ What is }\\
&\cos(90^{\circ}-x^{\circ})?\end{aligned}\)

\(\text{If }\sin x^{\circ}=a,\text{which of the following must be true for all values of }x?
\)

\(\text{Triangle }XYZ\text{ shown is a right triangle. Which of the following has the same value as }\sin x?\)

\(\text{Which of the following expressions is equivalent to }(\sin24^{\circ})(\cos66^{\circ})+(\cos24^{\circ})(\sin66^{\circ})?\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~2\cos (90-a)°\cdot\cos b°+\sin (a+y)°\cdot\sin (90-b)°\\
&\text{In the given expression, }a\text{ and }b\text{ are constants such that }\sin a°=0.50,\\
&\cos b°=0.99,\text{ and }0 < b < 90.\text{ What is the value of the given expression}\\
&\text{when }y=0?\end{aligned}\)

\(\begin{aligned}&\text{In triangle }RST\text{ above, point }W\text{ (not shown) lies on }\overline{RT}.\text{ What is the value of}\\&\cos(\angle RSW)-\sin(\angle WST)?\end{aligned}\)

\(\begin{aligned}
&\text{In triangle }DEF,\text{ point }G\text{ (not shown) lies on }\overline{DE}.\text{ If the measure of }\angle DFG\text{ is }x^{\circ}\text{, and}\\&\text{the measure of }\angle GFE\text{ is }y^{\circ}\text{, what is the value of }\cos x^{\circ}-\sin y^{\circ}?\end{aligned}\)

\(\begin{aligned}&\text{The angles shown above are acute and }\sin(a^\circ)=\cos(b^\circ).\text{ If }a=4k-22\text{ and }b=6k-13,\\
&\text{what is the value of }k?\end{aligned}\)

\(\begin{aligned}
&\text{For two acute angles }\angle Q\text{ and }\angle R,~\cos Q=\sin R.\text{ The measures, in degrees, of }\\
&\angle Q\text{ and }\angle R\text{ are }x+21\text{ and }4x+9\text{, respectively. What is the value of }x?\end{aligned}\)

\(\text{In the figure above, triangle }ABC\text{ is similar to triangle }DEF.\text{ What is the value of }\cos(E)?\)

\(\begin{aligned}
&\text{Triangle }FGH\text{ is similar to triangle }JKL\text{, where angle }F\text{ corresponds to angle }J\text{ and}\\
&\text{angles }G\text{ and }K\text{ are right angles. If} \sin (F)=\frac{308}{317}\text{, what is the value of }\sin (J)?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~RS=20\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ST=48\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~TR=52\\
&\text{The side lengths of right triangle }RST\text{ are given. Triangle }RST\text{ is similar to triangle }UVW,\\
&\text{where }S\text{ corresponds to }V\text{ and }T\text{ corresponds to }W\text{. What is the value of }\tan W?\end{aligned}\)

\(\begin{aligned}
&\text{In triangle }ABC,\text{the measure of }\angle B\text{ is }90^{\circ},BC=16\text{, and }AC=20.\text{ Triangle }DEF\\
&\text{is similar to triangle }ABC\text{, where vertices }D, E,\text{ and }F\text{ correspond to vertices }A,B,\\
&\text{and }C\text{, respectively, and each side of triangle }DEF\text{ is }\frac{1}{3}\text{ the length of the }\\
&\text{corresponding side of triangle }ABC\text{. What is the value of}\sin F?\end{aligned}\)

\(\begin{aligned}
&\text{Triangle }ABC\text{ is similar to triangle }DEF\text{, where angle }A\text{ corresponds to angle }D\text{ and}\\
&\text{angles }C\text{ and }F\text{ are right angles. The length }\overline{AB}\text{ is }2.4\text{ times the length of }\overline{DE}.\text{ If}\\
&\tan A=\frac{21}{20}\text{, what is the value of }\sin D?\end{aligned}\)

\(\text{In the figure above,}\tan B=\frac{3}{4}.\text{ If }BC=15\text{ and }DA=4\text{, what is the length of }\overline{DE}?\)

\(\text{In the figure above, }CD=5\text{ and }AD=8.\text{ What is the value of the cosine of }\angle AEB?\)

\(\begin{aligned}&\text{In the figure, }\overline{AD}\text{ and }\overline{BC}\text{ intersect at point }D.\text{ The measure of }\angle ABD\text{ is }w{°}\text{, the measure}\\
&\text{of }\angle CAD\text{ is }z{°},\text{ and }AD=178.\text{ If }\cos w{°}=\frac{9}{41},\text{what is the value of }\tan z{°}?\end{aligned}\)

\(\text{In triangle }QRS\text{ shown, }QR < RS.\text{ Which expression represents the length of }\overline {QS}?\)

\(\begin{aligned}
&\text{In triangle }XYZ,\text{the measure of angle }X\text{ is }90°.\text{Point }W\text{ lies on segment }YZ, \\
&\text{and segment }WX\text{ is perpendicular to segment }YZ.\text{ The length of segment }WY\\
&\text{is 572, and the length of segment }WX\text{ is 429. What is the value of }\tan Z?\end{aligned}\)