MathTV Logo TOPICS
 
CHOOSE A TOPIC
MathTV Logo

Search Results
Math Topics

Trigonometry
Identities and Formulas

1
Proving Identities

Problem  1

Prove \(\sin\theta\cot\theta=\cos\theta\)

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  2

Prove \(\tan x+\cos x=\sin x(\sec x+\cot x)\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  3

Prove \(\displaystyle\frac{\cos^4t-\sin^4t}{\cos^2t}=1-\tan^2t\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  4

Prove \(1+\cos\theta=\displaystyle\frac{\sin^2\theta}{1-\cos\theta}\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  5

Prove \(\tan x+\cot x=\sec x\csc x\).

Choose instructor to watch:
Stefanie cc
Stefanie
Adam cc
Adam
Gordon cc espanol spanish
Gordon
Problem  6

Prove \(\displaystyle\frac{\sin\alpha}{1+\cos\alpha}+\displaystyle\frac{1+\cos\alpha}{\sin\alpha}=2\csc\alpha\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  7

Prove \(\displaystyle\frac{1+\sin t}{\cos t}=\displaystyle\frac{\cos t}{1-\sin t}\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  8

Show that \(\cot^2\theta+\cos^2\theta=\cot^2\theta\cos^2\theta\) is not an identity by finding a counterexample.

Choose instructor to watch:
Julieta cc
Julieta
Stefanie cc
Stefanie
Adam cc
Adam
Julieta espanol spanish
Julieta
Mini Lecture

Mini Lecture
Prove.

  1. \(\cos\theta\tan\theta=\sin\theta\)

  2. \(\cos x(\csc x+\tan x)=\cot x+\sin x\)

  3. \(\displaystyle\frac{\cos^4t-\sin^4t}{\sin^2t}=\cot^2t-1\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

2
Sum and Difference Formulas

Problem  1

Find the exact value for \(\cos 75^\circ\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  2

Show that \(\cos(x+2\pi)=\cos x\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  3

Write \(\cos 3x\cos 2x-\sin 3x\sin 2x\) as a single cosine.

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  4

Show that \(\cos(90^\circ -A)=\sin A\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  5

Find the exact value of \(\sin \left(\displaystyle\frac{\pi}{12}\right)\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  6

If \(\sin A=\frac{3}{5}\) with \(A\) in QI and \(\cos B=-\frac{5}{13}\) with \(B\) in QIII, find \(\sin(A+B)\), \(\cos(A+B)\), and \(\tan(A+B)\).

Choose instructor to watch:
Stefanie cc
Stefanie
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  7

If \(\sin A=\frac{3}{5}\) with \(A\) in QI and \(\cos B=-\frac{5}{13}\) with \(B\) in QIII, find \(\tan(A+B)\) by using this formula \[\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\]

Choose instructor to watch:
Stefanie cc
Stefanie
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  8
Choose instructor to watch:
Adam cc
Adam
Julieta espanol spanish
Julieta
Mini Lecture

Mini Lecture

  1. Find the exact value of \(\sin 15^\circ\).

  2. Prove: \(\sin(90^\circ +\theta)=\cos\theta\)

  3. Graph one cycle: \(y=2\left(\sin x\cos\frac{\pi}{3}-\cos x\sin\frac{\pi}{3}-\cos x\sin\frac{\pi}{3}\right)\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

3
Double-Angle Formulas

Problem  1

If \(\sin A=\displaystyle\frac{3}{5}\) with \(A\) in QII, find \(\sin 2A\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  2

Prove \(\left(\sin\theta+\cos\theta\right)^2=1+\sin 2\theta\).

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Cynthia cc espanol spanish
Cynthia
Problem  3

If \(\sin A=\displaystyle\frac{1}{\sqrt{5}}\), find \(\cos 2A\).

Choose instructor to watch:
Stefanie cc
Stefanie
Aaron cc
Aaron
Cynthia cc espanol spanish
Cynthia
Problem  4

Prove \(\sin 2x=\displaystyle\frac{2\cot x}{\left(1+\cot^2x\right)}\).

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Julieta espanol spanish
Julieta
Problem  5

Prove \(\cos 4x=8\cos^4x-8\cos^2x+1\).

Choose instructor to watch:
Stefanie cc
Stefanie
CJ cc
CJ
Julieta espanol spanish
Julieta
Problem  6

Prove \(\tan\theta=\displaystyle\frac{1-\cos2\theta}{\sin2\theta}\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  7

Graph \(y=3-6\sin^2x\) from \(x=0\) to \(x=2\pi\).

Choose instructor to watch:
Adam cc
Adam
CJ cc
CJ
Julieta cc
Julieta
Julieta espanol spanish
Julieta
Problem  8
Choose instructor to watch:
Julieta cc
Julieta
Adam cc
Adam
Julieta espanol spanish
Julieta
Mini Lecture

Mini Lecture

  1. If \(\cos x=\displaystyle\frac{1}{\sqrt{10}}\) with \(x\in\) QIV, find \(\sin 2x\).

  2. Graph \(y=4-8\sin^2x\) from \(0\leq x\leq 2\pi\)

  3. Verify \(\sin 60^\circ=2\sin30^\circ\cos30^\circ\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

4
Half-Angle Formulas

Problem  1

If \(\cos A=\frac{3}{5}\) with \(270^\circ<A<360^\circ\), find \(\sin\frac{A}{2}\), \(\cos\frac{A}{2}\), and \(\tan\frac{A}{2}\).

Choose instructor to watch:
Stefanie cc
Stefanie
Gordon cc
Gordon
Gordon cc espanol spanish
Gordon
Problem  2

If \(\sin A=-\frac{12}{13}\) with \(180^\circ<A<270^\circ\), find the six trigonometric functions of \(\displaystyle\frac{A}{2}\).

Choose instructor to watch:
Adam cc
Adam
Julieta cc
Julieta
Julieta espanol spanish
Julieta
Problem  3

Find \(\tan15^\circ\).

Choose instructor to watch:
Adam cc
Adam
Julieta cc
Julieta
Julieta espanol spanish
Julieta
Problem  4

Prove \(\sin^2\displaystyle\frac{x}{2}=\displaystyle\frac{\tan x-\sin x}{2\tan x}\).

Choose instructor to watch:
Adam cc
Adam
Julieta cc
Julieta
Julieta espanol spanish
Julieta
Mini Lecture

Mini Lecture
Let \(\sin A=\displaystyle\frac{4}{5}\) with \(A\) in QII. Find.

  1. \(\sin\displaystyle\frac{A}{2}\)

  2. \(\cos 2A\)

  3. \(\sec 2A\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague

5
Additional Identities

Problem  1

Evaluate \(\sin\left(\arcsin\displaystyle\frac{3}{5}+\arctan 2\right)\) without using a calculator.

Choose instructor to watch:
Adam cc
Adam
Julieta cc
Julieta
Julieta espanol spanish
Julieta
Problem  2

Write \(\sin\left(2\tan^{-1}x\right)\) as an equivalent expression involving only \(x\). (Assume \(x\) is positive).

Choose instructor to watch:
Julieta cc
Julieta
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  3

Verify the product formula \((3)\) for \(A=30^\circ\) and \(B=120^\circ\).

Choose instructor to watch:
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  4

Write \(10\cos5x\sin3x\) as a sum or difference.

Choose instructor to watch:
Julieta cc
Julieta
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  5

Verify sum formula \((7)\) for \(\alpha=30^\circ\) and \(\beta=90^\circ\).

Choose instructor to watch:
Adam cc
Adam
Julieta espanol spanish
Julieta
Problem  6

Verify the identity \(-\tan x=\displaystyle\frac{\cos 3x-\cos x}{\sin 3x+\sin x}\).

Choose instructor to watch:
Julieta cc
Julieta
Adam cc
Adam
Julieta espanol spanish
Julieta
Mini Lecture

Mini Lecture
Evaluate.

  1. \(\sin\left(\arcsin\displaystyle\frac{3}{5}-\arctan2\right)\)

  2. \(\tan\left(\sin^{-1}x\right)\)

  3. Rewrite as a product: \(\sin7x+\sin3x\)

Choose instructor to watch:
Mr. McKeague cc
Mr. McKeague