Trigonometric Functions Review

Unit IN Review Notes

Trigonometric Functions

• Definition: Functions that relate angles to the ratios of sides in right triangles.

• Common Functions: sin (sine), cos (cosine), tan (tangent), csc (cosecant), sec (secant), cot (cotangent).

Key Trigonometric Identities

• Sum and Difference Formulas:

• ( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta )

• ( \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta )

• ( \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta} )

• ( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta )

• ( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta )

• ( \tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \tan\beta} )

Evaluating Trigonometric Functions

• For angles specified in radians and degree (e.g., 0, (\frac{\pi}{2}), (\pi), (\frac{3\pi}{2})), use unit circle or angle reduction techniques.

• Special Angles & Values:

• ( \sin 0 = 0, \sin \frac{\pi}{6} = \frac{1}{2}, \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} )

• ( \cos 0 = 1, \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \cos \frac{\pi}{3} = rac{1}{2} )

• ( \tan \frac{\pi}{4} = 1 )

Example Problems

• Simplifying Trigonometric Expressions:

• Example: Find (\sin(\alpha + \beta)): Substitute values for (\sin\alpha), (\cos\alpha), (\sin\beta), (\cos\beta) based on given angles.

• Finding Trigonometric Values:

• Example: Given (\sin x = \frac{3}{5}), find (\cos x): Use the Pythagorean identity (\sin^2 x + \cos^2 x = 1): (\cos x = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}).

Additional Identities

• Double Angle Formulas:

• ( \sin(2x) = 2\sin x\cos x )

• ( \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x )

• ( \tan(2x) = \frac{2\tan x}{1 - \tan^2 x} )

Calculus of Trigonometric Functions

• Differentiation and Integration of trigonometric functions require knowledge of derivatives:

• ( \frac{d}{dx}\sin x = \cos x, \frac{d}{dx}\cos x = -\sin x, \frac{d}{dx}\tan x = \sec^2 x )

• The integration often involves using identities or substitution techniques.

Conclusion

• Mastery of trigonometric identities and values, along with simplification and evaluation techniques, is essential for solving problems in calculus and other advanced mathematics topics. Practice with various angle measures and identities reinforces comprehension of these fundamental concepts.

Practice Problems for Trigonometric Functions

1. Evaluate the following trigonometric functions:
   a. ( \sin(\frac{\pi}{4}) )
   b. ( \cos(\frac{\pi}{3}) )
   c. ( \tan(\frac{3\pi}{4}) )

2. Use the sum and difference formulas to find:
   a. ( \sin(30^{\circ} + 45^{\circ}) )
   b. ( \cos(60^{\circ} - 30^{\circ}) )

3. Simplifying Trigonometric Expressions:
   a. Simplify ( \sin(\alpha + \beta) ) using the values ( \sin \alpha = \frac{1}{2} ) and ( \cos \beta = \frac{\sqrt{3}}{2} ).
   b. Simplify ( \tan(\alpha - \beta) ) using ( \tan \alpha = 1 ) and ( \tan \beta = \frac{\sqrt{3}}{3} ).

4. Finding Trigonometric Values:
   Given ( \sin x = \frac{4}{5} ), find ( \cos x ) and ( \tan x ).

5. Using Double Angle Formulas:
   a. Find ( \sin(2x) ) if ( \sin x = \frac{1}{2} ).
   b. Find ( \cos(2x) ) if ( \cos x = \frac{\sqrt{3}}{2} ).

6. Differentiate the following functions:
   a. ( y = \sin x )
   b. ( y = \cos(\frac{\pi}{4} x) )

7. Integrate the following functions:
   a. ( \int\sin x \, dx )
   b. ( \int\sec^2 x \, dx )

Practice Problems for Trigonometric Functions

1. Evaluate the following trigonometric functions:
   a. ( \sin(\frac{\pi}{4}) )
   b. ( \cos(\frac{\pi}{3}) )
   c. ( \tan(\frac{3\pi}{4}) )

2. Use the sum and difference formulas to find:
   a. ( \sin(30^{\circ} + 45^{\circ}) )
   b. ( \cos(60^{\circ} - 30^{\circ}) )

3. Simplifying Trigonometric Expressions:
   a. Simplify ( \sin(\alpha + \beta) ) using the values ( \sin \alpha = \frac{1}{2} ) and ( \cos \beta = \frac{\sqrt{3}}{2} ).
   b. Simplify ( \tan(\alpha - \beta) ) using ( \tan \alpha = 1 ) and ( \tan \beta = \frac{\sqrt{3}}{3} ).

4. Finding Trigonometric Values:
   Given ( \sin x = \frac{4}{5} ), find ( \cos x ) and ( \tan x ).

5. Using Double Angle Formulas:
   a. Find ( \sin(2x) ) if ( \sin x = \frac{1}{2} ).
   b. Find ( \cos(2x) ) if ( \cos x = \frac{\sqrt{3}}{2} ).

6. Differentiate the following functions:
   a. ( y = \sin x )
   b. ( y = \cos(\frac{\pi}{4} x) )

7. Integrate the following functions:
   a. ( \int\sin x \, dx )
   b. ( \int\sec^2 x \, dx )