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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

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Chapter 6: Trigonometry 6.5: Basic Trigonometric Identities

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  1. Chapter 6: Trigonometry6.5: Basic Trigonometric Identities Essential Question:What is the Pythagorean Identity? How can it be used to find other trigonometric identities?

  2. 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”

  3. 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.

  4. 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

  5. 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. 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 = ?

  7. 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

  8. 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

  9. 6.5: Basic Trigonometric Identities • Assignment • Page 460 • 1 – 25, odd problems • Show work

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