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Understanding Trigonometric Ratios in Right Triangles

In this lesson, students will learn to identify and calculate trigonometric ratios using right triangles. The essential concepts include defining sine, cosine, and tangent through the relationships of triangle sides (opposite, adjacent, and hypotenuse). Emphasis will be placed on using these ratios to solve practical problems involving right triangles. Students will also utilize calculators for trigonometric calculations, ensuring they switch from radians to degrees when necessary. Practice problems will reinforce these skills.

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Understanding Trigonometric Ratios in Right Triangles

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  1. Trigonometry Notes 6 – Section 8.4

  2. Essential Learnings • Students will understand and be able to find trigonometric ratios using right triangles. • Students will be able to use trigonometric ratios to solve problems involving right triangles.

  3. Vocabulary • Trigonometry comes from two Greek terms trigon meaning triangle, and metron meaning measure. • Trigonometric ratio – a ratio of the lengths of two sides of a right triangle.

  4. Right Triangle Trigonometry Given a right triangle ABC: The side opposite ∠A is a The side opposite ∠B is b C is the right angle Side c is the hypotenuse B c a C A b

  5. Trigonometric Ratios Three trigonometric ratios: sine, cosine, and tangent. SOHCAHTOA SOH – Sine: Opposite over Hypotenuse CAH – Cosine: Adjacent over Hypotenuse TOA – Tangent: Opposite over Adjacent

  6. Trigonometric Ratios Sine abbreviated sin: B c a C A b

  7. Trigonometric Ratios Cosine abbreviated cos: B c a C A b

  8. Trigonometric Ratios Tangent abbreviated tan: B c a C A b

  9. Example 1 (1-6) Express each ratio as a reduced fraction and as a decimal to the nearest hundredth. sin A = cos A = tan B = A 8 10 C 6 B

  10. Calculators & Nspires • Enter “Sin 30” into your calculator. Didyou get .5 or ½ ? You must change your settings from radians to degrees. On the Nspire Go to Home, Settings (5), Settings (2), General (1) Tab to Angle: Change setting to Degree (from radian) Tab to Make Default (at bottom of screen) and press enter.

  11. Example 2 (8-10, 28-33) Find x. Round to the nearest hundredth. 7 x 36⁰

  12. Example 3 (8-10, 28-35) Find x. Round to the nearest hundredth. x 63⁰ 12

  13. Class Problem Find x. Round to the nearest hundredth. 32 41⁰ x

  14. Example 4 (34 & 35) Your cat is stuck in a tree. You lean your 18 foot ladder against the tree where the cat is and the angle created by the ladder and the ground is 65⁰. How high is your cat?

  15. Assignment Pages 567 – 568: 1 – 11, 28 – 35 Math’s Mate 2-6 Unit Study Guide 5 Keep working on it

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