Trigonometric Identities (Chapter 3.3)

Practice flashcards covering key concepts and examples related to trigonometric identities and their establishment.

What are trigonometric identities?

Equations involving trigonometric functions that are true for all allowable values of the variable.

What is the Pythagorean identity involving sine and cosine?

Sine squared theta plus cosine squared theta equals one (\sin^2(\theta) + \cos^2(\theta) = 1).

What trigonometric function can be used to express tangent in terms of sine and cosine?

Tangent of theta equals sine of theta over cosine of theta (\tan(\theta) = \sin(\theta)/\cos(\theta)).

What is the reciprocal identity for cosecant?

Cosecant of theta is one over sine of theta (\csc(\theta) = 1/\sin(\theta)).

What is the reciprocal identity for secant?

Secant of theta is one over cosine of theta (\sec(\theta) = 1/\cos(\theta)).

What is the reciprocal identity for cotangent?

Cotangent of theta is one over tangent of theta (\cot(\theta) = 1/\tan(\theta)).

What is the Pythagorean identity involving tangent and secant?

Tangent squared theta plus one equals secant squared theta ($\tan^2(\theta) + 1 = \sec^2(\theta)).

What is the Pythagorean identity involving cotangent and cosecant?

One plus cotangent squared theta equals cosecant squared theta (1 + \cot^2(\theta) = \csc^2(\theta)).

What is the even identity for cosine?

Cosine of negative theta equals cosine of theta (\cos(-\theta) = \cos(\theta)).

What is the odd identity for sine?

Sine of negative theta equals negative sine of theta ($\sin(-\theta) = -\sin(\theta)).

What is the odd identity for tangent?

Tangent of negative theta equals negative tangent of theta ($\tan(-\theta) = -\tan(\theta)).

How can you derive the other two Pythagorean identities from \sin^2(\theta) + \cos^2(\theta) = 1?

Divide all terms by \cos^2(\theta) to get \tan^2(\theta) + 1 = \sec^2(\theta). Divide all terms by \sin^2(\theta) to get 1 + \cot^2(\theta) = \csc^2(\theta).

What is sine (sin) in a right triangle?

Opposite side over Hypotenuse (\sin(\theta) = \text{Opposite}/\text{Hypotenuse}).

What is cosine (cos) in a right triangle?

Adjacent side over Hypotenuse (\cos(\theta) = \text{Adjacent}/\text{Hypotenuse}).

What is tangent (tan) in a right triangle?

Opposite side over Adjacent side ($\tan(\theta) = \text{Opposite}/\text{Adjacent}).

What is cosecant (csc) in a right triangle?

Hypotenuse over Opposite side ($\csc(\theta) = \text{Hypotenuse}/\text{Opposite}).

What is secant (sec) in a right triangle?

Hypotenuse over Adjacent side ($\sec(\theta) = \text{Hypotenuse}/\text{Adjacent}).

What is cotangent (cot) in a right triangle?

Adjacent side over Opposite side ($\cot(\theta) = \text{Adjacent}/\text{Opposite}).

What mnemonic helps remember the basic trigonometric ratios in a right triangle?

SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

What is the cofunction identity for sine?

Sine of (pi/2 - theta) equals cosine of theta ($\sin(\frac{\pi}{2} - \theta) = \cos(\theta)).

What is the cofunction identity for cosine?

Cosine of (pi/2 - theta) equals sine of theta ($\cos(\frac{\pi}{2} - \theta) = \sin(\theta)).

What is the cofunction identity for tangent?

Tangent of (pi/2 - theta) equals cotangent of theta ($\tan(\frac{\pi}{2} - \theta) = \cot(\theta)).

What is the cofunction identity for cosecant?

\nCosecant of (pi/2 - theta) equals secant of theta ($\csc(\frac{\pi}{2} - \theta) = \sec(\theta)).

What is the cofunction identity for secant?

Secant of (pi/2 - theta) equals cosecant of theta ($\sec(\frac{\pi}{2} - \theta) = \csc(\theta)).

What is the cofunction identity for cotangent?

Cotangent of (pi/2 - theta) equals tangent of theta ($\cot(\frac{\pi}{2} - \theta) = \tan(\theta)).

What is the sine sum identity?

Sine of (A + B) equals sine A cosine B plus cosine A sine B ($\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)).

What is the sine difference identity?

Sine of (A - B) equals sine A cosine B minus cosine A sine B ($\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)).

What is the cosine sum identity?

Cosine of (A + B) equals cosine A cosine B minus sine A sine B ($\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)).

What is the cosine difference identity?

Cosine of (A - B) equals cosine A cosine B plus sine A sine B ($\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)).

What is the tangent sum identity?

Tangent of (A + B) equals (tangent A plus tangent B) over (1 minus tangent A tangent B) ($\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}).

What is the tangent difference identity?

Tangent of (A - B) equals (tangent A minus tangent B) over (1 plus tangent A tangent B) ($\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}).

What is the double angle identity for sine?

Sine of 2 theta equals 2 sine theta cosine theta ($\sin(2\theta) = 2\sin(\theta)\cos(\theta)).

What is the double angle identity for cosine (form 1)?

Cosine of 2 theta equals cosine squared theta minus sine squared theta ($\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)).

What is the double angle identity for cosine (form 2)?

Cosine of 2 theta equals 1 minus 2 sine squared theta ($\cos(2\theta) = 1 - 2\sin^2(\theta)).

What is the double angle identity for cosine (form 3)?

Cosine of 2 theta equals 2 cosine squared theta minus 1 ($\cos(2\theta) = 2\cos^2(\theta) - 1).

What is the double angle identity for tangent?

Tangent of 2 theta equals (2 tangent theta) over (1 minus tangent squared theta) ($\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}).

What is the half angle identity for sine?

Sine of theta over 2 equals plus or minus the square root of ((1 minus cosine theta) over 2) ($\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}).

What is the half angle identity for cosine?

Cosine of theta over 2 equals plus or minus the square root of ((1 plus cosine theta) over 2) ($\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}).

What is the half angle identity for tangent (form 1)?

Tangent of theta over 2 equals plus or minus the square root of ((1 minus cosine theta) over (1 plus cosine theta)) ($\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}).

What is the half angle identity for tangent (form 2)?

Tangent of theta over 2 equals (1 minus cosine theta) over sine theta ($\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)}).

What is the half angle identity for tangent (form 3)?

Tangent of theta over 2 equals sine theta over (1 plus cosine theta) ($\tan(\frac{\theta}{2}) = \frac{\sin(\theta)}{1 + \cos(\theta)}).

What is the product-to-sum identity for \sin(A)\cos(B)?

\sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] .

What is the product-to-sum identity for \cos(A)\sin(B)?

\cos(A)\sin(B) = \frac{1}{2}[\sin(A+B) - \sin(A-B)] .

What is the product-to-sum identity for \cos(A)\cos(B)?

\cos(A)\cos(B) = \frac{1}{2}[\cos(A+B) + \cos(A-B)] .

What is the product-to-sum identity for \sin(A)\sin(B)?

\sin(A)\sin(B) = \frac{1}{2}[\cos(A-B) - \cos(A+B)] .

What is the sum-to-product identity for \sin(A) + \sin(B)?

\sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) .

What is the sum-to-product identity for \sin(A) - \sin(B)?

\sin(A) - \sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) .

What is the sum-to-product identity for \cos(A) + \cos(B)?

\cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) .

What is the sum-to-product identity for \cos(A) - \cos(B)?

\cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) .

Simplify the expression: (1 - \sin^2(\theta))\sec^2(\theta)

(1 - \sin^2(\theta))\sec^2(\theta) = \cos^2(\theta)\sec^2(\theta) (using Pythagorean identity \sin^2(\theta) + \cos^2(\theta) = 1)
= \cos^2(\theta) \cdot \frac{1}{\cos^2(\theta)} (using reciprocal identity for secant)
= 1

If \sin(\theta) = 3/5 and \theta is in Quadrant II, find the exact value of \cos(-\theta)

Using the Pythagorean identity, \sin^2(\theta) + \cos^2(\theta) = 1.
(3/5)^2 + \cos^2(\theta) = 1
9/25 + \cos^2(\theta) = 1
\cos^2(\theta) = 1 - 9/25 = 16/25
Since \theta is in Quadrant II, \cos(\theta) is negative.
\cos(\theta) = -\sqrt{16/25} = -4/5
Using the even identity for cosine, \cos(-\theta) = \cos(\theta).
Therefore, \cos(-\theta) = -4/5

Evaluate the exact value of \sin(75^\circ) using a sum identity.

\sin(75^\circ) = \sin(45^\circ + 30^\circ)
Using the sine sum identity, \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B).
\sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ).\sin(30^\circ)
= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})
= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}

If \cos(\theta) = 5/13 and \theta is an acute angle, find the exact value of \cos(2\theta)

Using the double angle identity for cosine: \cos(2\theta) = 2\cos^2(\theta) - 1 (form 3)
Substituting the given value:
\cos(2\theta) = 2(5/13)^2 - 1
= 2(25/169) - 1
= 50/169 - 1
= 50/169 - 169/169
= -119/169

Find the exact value of \tan(15^\circ) using a half-angle identity.

\tan(15^\circ) = \tan(\frac{30^\circ}{2}).
Using the half-angle identity for tangent: \tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)}.
Let \theta = 30^\circ.
\tan(15^\circ) = \frac{1 - \cos(30^\circ)}{\sin(30^\circ)}
= \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}}
= \frac{\frac{2 - \sqrt{3}}{2}}{\frac{1}{2}}
= 2 - \sqrt{3}