Lesson
Notes
Questions
Verify the following.
\(\textbf{1)}\) \( \csc ^2 {x} (1-\cos ^2 {x}) = 1 \)
\(\textbf{2)}\) \( \tan ^2 {x} (\csc ^2 {x}-1) = 1 \)
\(\textbf{3)}\) \( \sec{x} – \cos{x} =\frac{\tan^2{x}}{\sec{x} } \)
\(\textbf{4)}\) \( \sin ^2 {x} -\cos ^2 {x} = 1-2\cos ^2 {x} \)
\(\textbf{5)}\) \( -\sec{x}\tan{x} = \frac{\csc{x}}{1-\csc^2{x}} \)
\(\textbf{6)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)
\(\textbf{7)}\) \( \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})^2-1}=\frac{1}{2} \)
\(\textbf{8)}\) \( \sin{x} + \csc{x}\cos^2{x} = \csc{x} \)
\(\textbf{9)}\) \( \cot ^2 {x} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \)
\(\textbf{10)}\) \( (\sin{x}+\cos{x})^4 = (1+2\sin{x}\cos{x})^2 \)
\(\textbf{11)}\) \( \frac{\sec{x}}{\sec{x}-\cos{x}} = \csc^2{x} \)
\(\textbf{12)}\) \( (\csc{x}-\cot{x})(\cos{x}+1)=\sin{x} \)
\(\textbf{13)}\) \( 2\sin ^2 {x} -1=1-2\cos ^2 {x} \)
\(\textbf{14)}\) \( \sin ^4 {x} -\cos ^4 {x} =\sin ^2 {x} -\cos ^2 {x} \)
\(\textbf{15)}\) \( (1-\sin ^2 {x} )(1+\sin ^2 {x} )=1-\sin ^4 {x} \)
\(\textbf{16)}\) \( \frac{\tan{x}+\cot{x}}{\sin{x}\cos{x}} =\sec^2{x}+\csc^2{x} \)
\(\textbf{17)}\) \( \frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos^2{x} -\sin^2{x} \)
\(\textbf{18)}\) \( \frac{1+\sec^2{x}}{1+\tan^2{x}}=1+\cos^2{x} \)
\(\textbf{19)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)
\(\textbf{20)}\) \( \sec ^4 {x} -\tan ^4 {x} =\sec ^2 {x} +\tan ^2 {x} \)
See Related Pages\(\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)
\(\bullet\text{ Andymath Homepage}\)
