Chapter 6 Trigonometry 6 5 Basic Trigonometric Identities

Chapter 6: Trigonometry 6. 5: Basic Trigonometric Identities Essential Question: What is the Pythagorean Identity? How can it be used to find other trigonometric identities?

6. 5: Basic Trigonometric Identities • Convert to radian mode for this section ▫ 2 nd → more • Parenthesis matter ▫ “cos (x + 3)” and “cos x + 3” yield different results �Why? ▫ (cos t)2 is written (on paper) as cos 2 t, because we’re squaring the result of the cosine function, not the “t”

6. 5: Basic Trigonometric Identities • Quotient Identities ▫ ▫ Example 1: Simplify the expression: tan t cos t ▫ The key for conversion is to get everything in terms of sin and cos. ▫

6. 5: Basic Trigonometric Identities • Reciprocal Identities ▫ We’ve seen these before ▫ ▫ ▫ Example 2: Reciprocal Identities �Given that sin t = 0. 28 and cos t = 0. 96. Find csc t and sec t � �

6. 5: Basic Trigonometric Identities • Pythagorean Identities ▫ You know the traditional, a 2 + b 2 = c 2 ▫ sin 2 t + cos 2 t = 1 �The other two identities can be derived from this equation ▫ tan 2 t + 1 = sec 2 t ▫ 1 + cot 2 t = csc 2 t ▫ Example 3: Using Pythagorean identities �Simplify the equation: tan 2 t cos 2 t + cos 2 t �Rewrite using just sin & cos �

6. 5: Basic Trigonometric Identities • Example 4: Finding all other values from one • The value for trigonometric function is given for 0 < t < π/2. Use quotient, reciprocal and Pythagorean identities to find the other values of the remaining five trigonometric functions. Round your answers to four decimal places. • sec t = 2. 5846 cos t = ? • csc t = ? sin t = ? • cot t = ? tan t = ?

6. 5: Basic Trigonometric Identities • sec t = 2. 5846 • csc t = ? • cot t = ? cos t = 0. 3869 sin t = ? tan t = 2. 3833 • cos can be found as it’s the reciprocal of sec • cos t = 1/sec t = 1/2. 5846 = 0. 3869 • • tan can be found with the Pythagorean theorem sec 2 t = tan 2 t + 1 2. 58462 – 1 = tan 2 t 2. 3833 = tan t

6. 5: Basic Trigonometric Identities • sec t = 2. 5846 • csc t = 1. 0850 • cot t = 0. 4199 cos t = 0. 3869 sin t = 0. 9217 tan t = 2. 3833 • cot can be found as it’s the reciprocal of tan • cot t = 1/tan t = 1/2. 3833 = 0. 4199 • • tan t = sin t / cos t 2. 3833 = sin t / 0. 3896 2. 3833 ● 0. 3896 = sin t 0. 9217 = sin t • csc can be found by taking the reciprocal of sin • csc t = 1/sin t = 1/0. 9217 = 1. 0850

6. 5: Basic Trigonometric Identities • Assignment ▫ Page 460 � 1 – 25, odd problems �Show work