Introduction to Trig Ratios and to Identity
Trig Ratios
\sin \theta = \dfrac{opp}{hyp}
\cos \theta = \dfrac{adj}{hyp}
\tan \theta = \dfrac{opp}{adj}
Reciprocal Identities
\csc \theta = \dfrac{1}{\sin \theta}
\sec \theta = \dfrac{1}{\cos \theta}
\cot \theta = \dfrac{1}{\tan \theta}
Pythagorean Identities
\sin^2 \theta + \cos^2 \theta = 1
Simplifying \cot x \sin x
ex Simplify cot \theta \times \sin \theta.
= \dfrac{\cos \theta}{\sin \theta} \times \sin \theta
= \cos \theta
Simplifying Trig Expression 1+\tan^2x
ex 1+\tan^2x
= 1+ (\dfrac{\sin \theta}{\cos \theta})^2
= 1 + \dfrac{\sin^2 \theta}{\cos^2 \theta}
= \dfrac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}
= \dfrac{1}{\cos^2 \theta}
= \sec^2 \theta
Simplifying Trig Expression \dfrac{1 + \sin \theta}{\cos^2 \theta}
ex Simplify \dfrac{1 + \sin \theta}{\cos^2 \theta}
= \dfrac{1 + \sin \theta}{1 - \sin^2 \theta}
= \dfrac{1 + \sin \theta}{(1 + \sin \theta)(1 - \sin \theta)}
= \dfrac{1}{1 - \sin \theta}
Simplifying Trig Expression
Introduction to Proving Identities
LS = RS
Prove that LS is the same as RS.
ex \dfrac{1}{\cos \theta} + \tan \theta = \dfrac{1 + \sin \theta}{\cos \theta}
Proving Identity with Relative Angles
Proving Identity ex2
ex Prove 2 \cos^2 \theta + \sin^2 \theta - 1 = \cos^2 \theta
Proving Identity ex3
ex Prove \tan \theta + \dfrac{1}{\tan \theta} = \dfrac{1}{\sin \theta \cos \theta}
Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.
\displaystyle
\text{cot }\theta = \frac{\text{cos }\theta}{\text{sin }\theta}
Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.
\displaystyle
\text{tan }\theta \text{ cos }\theta = \text{ sin }\theta
Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.
\displaystyle
\text{csc }\theta = \frac{1}{\text{sin }\theta}
Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.
\displaystyle
\text{cos }\theta \text{ sec }\theta = 1
Simplify each expression.
\displaystyle
(1 - \text{sin }\alpha)(1 + \text{sin }\alpha)
Simplify each expression.
\displaystyle
\frac{\text{tan }\alpha}{\text{sin }\alpha}
Simplify each expression.
\displaystyle
\text{cos}^{2}\alpha + \text{sin}^{2}\alpha
Simplify each expression.
\displaystyle
\text{cot }\alpha \text{ sin }\alpha
Factor each expression.
\displaystyle
1 - \text{cos}^{2}\theta
Factor each expression.
\displaystyle
\text{sin}^{2}\theta - \text{cos}^{2}\theta
Factor each expression.
\displaystyle
\text{sin}^{2}\theta - 2\text{sin }\theta + 1
Factor each expression.
\displaystyle
\text{cos }\theta - cos^{2}\theta
Prove that \frac{\cos^{2}\phi}{1 - \sin\phi} = 1 + \sin\phi, where \sin\phi \neq 1, by expressing \cos^{2}\phi in terms of \sin \phi.
Prove the identity. State any restrictions on the variables.
\displaystyle
\frac{\sin x}{\tan x} = \cos x
Prove the identity. State any restrictions on the variables.
\displaystyle
\frac{\tan\theta}{\cos\theta} = \frac{\sin\theta}{1 - \sin^{2}\theta}
Prove the identity. State any restrictions on the variables.
\displaystyle
\frac{1}{\cos\alpha} + \tan\alpha = \frac{1 + \sin\alpha}{\cos\alpha}
Prove the identity. State any restrictions on the variables.
\displaystyle
1 - \cos^{2}\theta = \sin\theta\cos\theta\tan\theta
Mark claimed that \displaystyle \frac{1}{\cot \theta} = \tan \theta is an identity. Nancy let \theta = 30^o and found that both sides of the equation worked out to 1 . She said that this proves that the equation is an identity. Is Nancy's reasoning correct? Explain.
Simplify each trigonometric expression.
\displaystyle
\sin \theta \cot \theta - \sin \theta \cos \theta
Simplify each trigonometric expression.
\displaystyle
\cos\theta(1 + \sec\theta)(\cos\theta - 1)
Simplify each trigonometric expression.
\displaystyle
(\sin x +\cos x)(\sin x -\cos x) + 2\cos^2x
Simplify each trigonometric expression.
\displaystyle
\frac{\csc^{2}\theta - 3\csc\theta + 2}{\csc^{2}\theta - 1}
Prove each identity. State any restrictions on the variables.
\dfrac{\sin^{2}\phi}{1 - \cos\phi} = 1 + \cos\phi
Prove each identity. State any restrictions on the variables.
\dfrac{\tan^{2}\alpha}{1 + \tan^{2}\alpha} = \sin^{2}\alpha
Prove each identity. State any restrictions on the variables.
\cos^{2}x = (1 - \sin x)(1 + \sin x)
Prove each identity. State any restrictions on the variables.
\sin^{2}\theta + 2\cos^{2}\theta - 1 = \cos^{2}\theta
Prove each identity. State any restrictions on the variables.
\sin^{4}\alpha - cos^{4}\alpha = \sin^{2}\alpha - \cos^{2}\alpha
Prove each identity. State any restrictions on the variables.
\tan\theta + \dfrac{1}{\tan\theta} = \dfrac{1}{\sin\theta\cos\theta}
Is \csc^{2}\theta + \sec^{2}\theta = 1 an identity? Prove that it is true or demonstrate why it is false.
Prove that \sin^{2}x(1 + \dfrac{1}{\tan^{2}x}) = 1, where \sin x \neq 0.
Prove the identity. State any restrictions on the variables.
\displaystyle
\frac{\sin^2x + 2\cos x - 1}{\sin^2 x +3\cos x -3} = \frac{\cos^2x + \cos x}{-\sin^2 x}
Prove the identity. State any restrictions on the variables.
\displaystyle
\sin^2x -\cos^2x -\tan^2x = \frac{2\sin^2x -2\sin^4 x -1}{1- \sin^2 x}
Show how you can create several new identities from the identity
\sin^2x + \cos^2 x=1 by adding, subtracting, multiplying, or dividing
both sides of the equation by the same expression.
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
(1 - \cos^{2}x)(1 - \tan^{2}x) = \frac{\sin^{2}x - 2\sin^{4}x} {1 - \sin^{2}x}
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
1 - 2\cos^{2}\phi = \sin^{4}\phi - \cos^{4}\phi
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
\displaystyle
\dfrac{\sin x\tan x}{\sin x + \tan x} = \sin x \tan x
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
\displaystyle
\frac{1 + 2\sin x \cos x}{\sin x + \cos x} = \sin x + \cos x
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
\displaystyle
\frac{1 -\cos x}{\sin x} = \frac{\sin x}{1 + \cos x}
- (a) Is this an identity? Justify your answers.
- (b) For those equations that are identities, state and restrictions on the variables.
\displaystyle
\frac{\sin x}{ 1 + \cos x} = \csc x - \cot x