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Understanding Inverse Trigonometric Functions: Sine, Cosine, and Tangent

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This lesson focuses on defining the inverse of the sine, cosine, and tangent functions through their respective arcsin, arccos, and arctan functions. Students will learn about the domains and ranges of these inverse functions. Key examples will include finding the exact values of specific expressions such as sin⁻¹(½), cos⁻¹(0), and tan⁻¹(1). This lesson is crucial for mastering inverse trigonometric functions necessary for higher mathematics. Practice problems will be assigned from the textbook.

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Understanding Inverse Trigonometric Functions: Sine, Cosine, and Tangent

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  1. Check Homework

  2. PRE Lesson 7.4: Inverse Trig Functions • Objectives: • To define the inverse of the sine, cosine, and tangent functions. • To understand the domain and range of the inverse trig functions.

  3. Inverse Sine

  4. Inverse Sine (arcsin) The inverse of the function sin is the function sin–1 defined by: for –1 ≤ x ≤ 1 and –π/2 ≤ y ≤π/2. sin y = x sin–1x = y

  5. Example 1. Find the exact value of the expression if it is defined: • sin–1 ½ • sin–1 (–½) • sin–1 (3/2)

  6. Inverse Cosine

  7. Inverse Cosine (arccos) The inverse cosine functionis the function cos–1with domain [–1, 1] and range [0, π], defined by: cos y = x cos–1 x = y

  8. Example 2. Find the exact value of the expression if it is defined: • cos–1 • cos–1 (0) • cos–1 (5/7)

  9. Inverse Tangent

  10. Inverse Tangent (arctan) The inverse tangent functionis the function tan–1with domain and range (–π/2, π/2) defined by: tan y = x tan–1 x = y

  11. Example 3. Find the exact value of the expression if it is defined: • tan–1 (1) • tan–1 ( ) • tan–1 (-20)

  12. Ex 4. Find the exact value of the expression, if it is defined. a.) b.) c.)

  13. Classwork:Book; pg. 557: 1,4,6,7,10,11,13,16,18, 21, 26,53,54

  14. Classwork:Book; pg. 557: 1,4,6,7,10,11,13,16,18, 21, 26,53,54

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