Double-Angle and Half-Angle Formulas

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Notes

Notes for Double-Angle Formula

Notes for Half-Angle Formula


Questions

\(\textbf{1)}\) \(\text{Find exact value of }\sin{165^{\circ}}\)


\(\textbf{2)}\) \(\text{Find exact value of }\cos{75^{\circ}}\)


\(\textbf{3)}\) \(\text{Find exact value of }\tan{67.5^{\circ}}\)


\(\textbf{4)}\) \(\text{Find exact value of }\cos{15^{\circ}}\)


\(\textbf{5)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\sin ⁡2\theta\) Link to Youtube Video Solving Question Number 5


\(\textbf{6)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\cos ⁡2\theta\) Link to Youtube Video Solving Question Number 6


\(\textbf{7)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\tan ⁡2\theta\)


\(\textbf{8)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\sin \frac{\theta}{2}\)


\(\textbf{9)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\cos \frac{\theta}{2}\)


\(\textbf{10)}\) \(\sin\theta=\frac{3}{5},\) and \(90^{\circ}\lt\theta\lt180^{\circ},\) find \(\tan \frac{\theta}{2}\)


\(\textbf{11)}\) \(\sin{A}=\frac{4}{5},\) and \(0^{\circ}\lt A \lt90^{\circ},\) find \(\tan{2A}\)


\(\textbf{12)}\) \(\sin{A}=\frac{4}{5},\) and \(90^{\circ}\lt A \lt180^{\circ},\) find \(\tan{2A}\)


Find each value.

\(\textbf{13)}\) \(2 \sin (75) \cos (75)⁡ \)


\(\textbf{14)}\) \(\cos^2(22.5)-\sin^2⁡(22.5)\)


Verify the Identities using Double Angle Formulas

\(\textbf{15)}\) \(\displaystyle\frac{\sin{2x}}{\sin{x}}-\frac{\cos{2x}}{\cos{x}}=\sec{x}\)


\(\textbf{16)}\) \(\displaystyle\frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos{2x}\)



See Related Pages\(\)

\(\bullet\text{ Trig Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Reference Angles\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for All 6 Trig Functions\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) Thumbnail for Law of Sines\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) Thumbnail for Area of SAS Triangles\(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) Thumbnail for Law of Cosines\(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) Thumbnail for Heron's Formula\(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Similar Right Triangles\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) Thumbnail for Graphing Trig Functions\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)


About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!