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 2nd → more Parenthesis matter “cos (x + 3)” and “cos x + 3” yield different results Why? (cos t)2 is written (on paper) as cos2 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, a2 + b2 = c2 sin2 t + cos2 t = 1 The other two identities can be derived from this equation tan2 t + 1 = sec2 t 1 + cot2 t = csc2 t Example 3: Using Pythagorean identities Simplify the equation: tan2 t cos2 t + cos2 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 cos t = 0.3869 csc t = ? sin t = ? cot 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 sec2 t = tan2 t + 1 2.58462 – 1 = tan2 t 2.3833 = tan t

6.5: Basic Trigonometric Identities sec t = 2.5846 cos t = 0.3869 csc t = 1.0850 sin t = 0.9217 cot t = 0.4199 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