Equations that incorporate elements of trigonometry and complex numbers are known as complex trigonometric equations. They are applied to determine unknown variables, utilizing the concepts of trigonometry and the attributes of complex numbers. Frequently, they involve the application of Euler's formula to streamline the computational process.
| Topic | Problem | Solution |
|---|---|---|
| None | Score: $1 / 2$ Penalty: none Question Solve the t… | The given equation is a quadratic equation in terms of \(\sin^2 x\). We can solve it by setting it … |
| None | If $\cot (x)=\frac{5}{11}$ (in Quadrant -1$)$, fi… | We are given that \(\cot (x)=\frac{5}{11}\) and we know that \(\cot (x)=\frac{\cos (x)}{\sin (x)}\)… |
| None | Solve $8 \sin ^{2}(w)-2 \sin (w)-3=0$ for all sol… | The given equation is a quadratic equation in terms of \(\sin(w)\). We can solve it by using the qu… |
| None | Solve exactly for $x$ for $720^{\circ} \leq x \le… | The given equation is in terms of trigonometric functions. We can simplify the equation by using th… |
| None | Use trigonometric identities to solve the trigono… | Given the trigonometric equation \(\sin (2 \theta)+\cos \theta=0\). |
| None | Find all solutions to the equation $11 \cos x=\fr… | Given the equation \(11 \cos x=\frac{1}{\cos x}\). This can be rewritten as \(11 \cos^2 x = 1\). |
| None | Find all solutions of the equation in the interva… | Since \(\sin x = \sqrt{1 - \cos^2 x}\), we get \(-\sqrt{1 - \cos^2 x} = -\cos^2 x - 1\). |
| None | Use the given information to find (a) $\sin (s+1)… | Given that \(\cos s = -\frac{3}{5}\) and \(\sin t = -\frac{12}{13}\), we need to find the values of… |
| None | Find the exact value of the function. $\tan \frac… | We are given that \(\tan \beta = \frac{\sqrt{5}}{2}\) and we need to find the exact value of \(\tan… |
| None | Find the five remaining trigonometic functions of… | Given that \(\sec \theta=\frac{8}{7}\), and \(\sin \theta<0\) |
| None | Solve the equation on the interval $[0,2 \pi)$. \… | By the double-angle formula, \(\sin 2x = 2 \sin x \cos x\), so \(\sin 2x = -\sqrt{3} \sin x\) becom… |
| None | Solve the equation for exact solutions over the i… | The given equation is \(2 \sqrt{3} \sin 2 \theta=-3\). |
| None | Solve the equation for exact solutions in the int… | The given equation is \(6 \sec ^{2} \theta \tan \theta=8 \tan \theta\). |
| None | Solve the equation for solutions over the interva… | Given the equation \(\csc ^{2} \theta-2 \cot \theta=0\), we need to solve for \(\theta\) over the i… |
| None | Solve the equation for solutions in the interval … | The given equation is a product of two factors equal to zero. This means that either of the factors… |
| None | Solve the equation for exact solutions over the i… | We can write the equation as \[3 \frac{\cos x}{\sin x} + 2 = 5.\] |
| None | Give the degree measure of $\theta$ if it exists.… | Given that \(\theta=\cos^{-1}(2)\) |
| None | Find the exact value of the real number $y$ if it… | Find the exact value of the real number \(y\) if it exists. Do not use a calculator. |
| None | Find the least positive value of $\theta$. \[ \co… | The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, the equation c… |
| None | ii) Solve $3 \csc ^{2} 2 \theta+5 \cot 2 \theta=3… | First, we rewrite the given equation using the reciprocal identities for cosecant and cotangent. Th… |
| None | Solve the equation in degrees for all exact solut… | Given the trigonometric equation \(1 - \sin 2 \theta = 3 \sin 2 \theta\). |
| None | Find all exact solutions on the interval $[0,2 \p… | Given the equation \(\sin ^{2}(x)-\cos ^{2}(x)-\sin (x)=0\). |
| None | Find the real zeros of the trigonometric function… | We are given the function \(f(x) = -\sin(2x) + \sin(x)\) and we need to find the real zeros of this… |
| None | 12. Without using a calculator, determine two ang… | Since the cosecant is the reciprocal of the sine function, we need to find two angles with a sine o… |
| None | a) Find the value of $m$ if $\sec 65^{\circ}=\ope… | Given that \(\sec 65^\circ = \operatorname{cosec}(2m - 15)^\circ\) |