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17 problems match “trigonometric identities”. Clear

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1. Cosine of 15 degrees ●○○○○ 10 ptsFind the value of $\cos 15^\circ$. Round to two decimal places.
2. Double-angle sine from given sine and cosine ●○○○○ 10 ptsIf $\theta$ is an angle in the first quadrant with $\sin(\theta) = \frac{3}{5}$ and $\cos(\theta) = \frac{4}{5}$, find $\sin(2\theta)$. Give your answer as a decimal.
3. Sum of inverse sines of sines ●●○○○ 20 ptsCompute $\sin^{-1} (\sin 3) + \sin^{-1} (\sin 4) + \sin^{-1} (\sin 5).$ All functions are in radians.
4. Triangle area from a dot product ●●○○○ 20 ptsIn triangle $ABC$, $AB=1$, $BC=3$, and $\overrightarrow{AB}\cdot\overrightarrow{BC}=-1$. Find the area of triangle $ABC$. Round to two decimal places.
5. Simplify a complex exponential ratio ●●○○○ 20 ptsLet $\theta$ be an angle such that $\tan \theta = 2$. Simplify $\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}$ and write the result in the form $a+bi$ with real numbers $a$ and $b$. Find $a$ and $b$.
6. Solutions of a trigonometric equation ●●○○○ 20 ptsFind the number of solutions to $\frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta} - \frac{1}{\tan^2 \theta} - \frac{1}{\cot^2 \theta} - \frac{1}{\sec^2 \theta} - \frac{1}{\csc^2 \theta} = -3$ in the interval $0 \le \theta \le 2\pi$.
7. Ratio of sine and cosine differences ●●○○○ 20 ptsEvaluate $\frac{\sin 35^{\circ} -\sin 25^{\circ}}{\cos 35^{\circ} -\cos 25^{\circ}}$. Round to two decimal places.
8. Trigonometric identities with complementary angles ●●○○○ 20 ptsEvaluate $(\cos^2 25^\circ + \cos^2 65^\circ) + \csc\theta \sec(90^\circ - \theta) - \cot\theta \tan(90^\circ - \theta)$.
9. Maximum of a trigonometric quadratic in cosine ●●○○○ 20 ptsFind the maximum value of the function $f(x)=\sin^{2}x+\sqrt{3}\cos x-\frac{3}{4}$ for $x\in\left[0,\frac{\pi}{2}\right]$.
10. Sums of squares on a unit circle ●●●○○ 50 ptsThree points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ lie on the unit circle $x^2+y^2=1$ and satisfy $x_1+x_2+x_3=0$ and $y_1+y_2+y_3=0$. Find $x_1^2+x_2^2+x_3^2$ and $y_1^2+y_2^2+y_3^2$.

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