In mathematics, fundamental identities serve as the critical formulas that establish a connection between varying mathematical values. They are the essence of trigonometry, aiding in the simplification of intricate expressions and the resolution of equations. Essential identities encompass reciprocal, quotient, Pythagorean, co-function, even/odd, and double/half angle identities. Acquiring a thorough understanding of these is an essential step towards tackling more sophisticated mathematical challenges.
| Topic | Problem | Solution |
|---|---|---|
| None | Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, … | We are given that \(\cos(A) = -\frac{21}{29}\) and that \(A\) is in the third quadrant. |
| None | Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, … | We are given that \(\cos(A) = -\frac{21}{29}\) and that \(A\) is in quadrant 3. |
| None | Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, … | Given that \(\cos(A) = -\frac{21}{29}\) and \(A\) is in quadrant 3 |
| None | Let $\cos A=-\frac{2}{\sqrt{13}}$ with $A$ in $Q … | Given that \(\cos A = -\frac{2}{\sqrt{13}}\) with A in Quadrant III, we are asked to find \(\cot (2… |
| None | Use identities to find values of the sine and cos… | We are given that \(\tan x = -2\) and \(\cos x < 0\). |
| None | Decide whether the statement is possible or impos… | Decide whether the statement is possible or impossible: \(\cos \alpha=-\frac{1}{2}\) and \(\sec \al… |
| None | Use an identity to write the expression as a sing… | Given the expression \(\frac{1}{6}-\frac{1}{3} \sin ^{2} 35.9^{\circ}\) |
| None | Use identities to fill in the blank. If $\tan \th… | The cotangent of an angle is the reciprocal of the tangent of that angle. |
| None | Use the appropriate reciprocal identity to find t… | We are given that \(\csc \theta = \frac{\sqrt{63}}{7}\) |
| None | Use a double-angle identity to find the exact val… | The given expression is in the form of a double angle identity for cosine. The double angle identit… |
| None | Find the expression that is equivalent to $\frac{… | The given expression is \(\frac{\tan \frac{3 \pi}{4}+\tan \frac{\pi}{6}}{1-\tan \frac{3 \pi}{4} \ta… |
| None | Find the exact value of the expression by using a… | Let's denote \(a = \sin 72^\circ\) and \(b = \cos 42^\circ\). Then we have \(\sin 72^\circ \cos 42^… |
| None | Use identities to correctly complete the sentence… | The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle… |