Trigonometric identities II

Learning Objectives Apply the sum and difference identities to find exact trigonometric values for non-standard angles. Utilize double-angle identities to simplify complex trigonometric expressions. Prove trigonometric identities that require the use of sum, difference, or double-angle formulas. Select the most appropriate form of the cosine double-angle identity to solve a problem efficiently. Solve trigonometric equations by substituting advanced identities to create solvable forms. Derive the double-angle identities from the sum identities. Ever wondered how to find the exact value of sin(15°) without a calculator? 🤔 These advanced identities are your secret key! This tutorial moves beyond the basic Pythagorean identities to explore the powerful sum, difference, and dou...

TermDefinitionExample Sum IdentityA formula that expresses a trigonometric function of a sum of two angles (A + B) in terms of trigonometric functions of the individual angles A and B.The sum identity for sine is \(\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)\). To find \(\sin(75°)\), you can use \(\sin(45°+30°)\). Difference IdentityA formula that expresses a trigonometric function of a difference of two angles (A - B) in terms of trigonometric functions of the individual angles A and B.The difference identity for cosine is \(\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)\). To find \(\cos(15°)\), you can use \(\cos(45°-30°)\). Double-Angle IdentityA special case of the sum identity where the two angles are equal (A = B), expressing a trigonometric function of a doubled angle (2A) in terms o...

Sum and Difference Identities \(\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)\\) \(\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)\) \(\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}\) Use these to find exact values of angles that can be expressed as a sum or difference of special angles (like 30°, 45°, 60°). Note the sign change in the cosine identity. Double-Angle Identities \(\sin(2A) = 2\sin(A)\cos(A)\) \(\cos(2A) = \cos^2(A) - \sin^2(A) = 2\cos^2(A) - 1 = 1 - 2\sin^2(A)\) \(\tan(2A) = \frac{2\tan(A)}{1-\tan^2(A)}\) Use these to simplify expressions involving a doubled angle or to solve equations. The three forms for \(\cos(2A)\) are interchangeable; choose the one that best suits the problem.