Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define sin, cos, and tan for angles from 0° to 360°.
• Determine the signs of trig ratios in each quadrant.
• Use reference angles for angles greater than 90°.
• Calculate ratios of special angles without a calculator.

Lesson summary

This week, we delve into the world of Trigonometric Functions, specifically focusing on how to find trigonometric ratios of angles in the Cartesian plane (beyond acute angles) and how to determine the signs of these ratios in different quadrants. This is a crucial step in understanding trigonometry, as it allows us to work with angles of any size and apply trigonometric principles in more complex scenarios. Understanding trigonometry unlocks doors to fields like surveying, navigation, engineering, and even music! For example, cellphone tower placement relies heavily on trigonometric calculations to ensure optimal signal coverage across various terrains in South Africa.

Lesson notes

2.1 The Cartesian Plane and Trigonometric Ratios Recall the Cartesian plane, divided into four quadrants: Quadrant I: x > 0, y > 0 (Angles between 0° and 90°)

Quadrant II: x 0 (Angles between 90° and 180°)

Quadrant III: x 0, y 90°:** Determine the quadrant in which the angle lies. Determine the sign of the trigonometric ratio in that quadrant using CAST. Calculate the reference angle (θ'). Find the trigonometric ratio of the reference angle (sin θ', cos θ', or tan θ'). Apply the sign determined in step 2 to the value obtained in step 4. 2.4 Special Angles (0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°) These angles occur frequently, and it's beneficial to know their trigonometric ratios without a calculator. Consider using the unit circle and special right triangles (30-60-90 and 45-45-90) to derive these values. | Angle (θ) | sin θ | cos θ | tan θ | |---|---|---|---| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 or √3/3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | Undefined | | 180° | 0 | -1 | 0 | | 270° | -1 | 0 | Undefined | | 360° | 0 | 1 | 0 |

Worked example

Example 1: Find sin 150°.

150° lies in Quadrant II.

Sine is positive in Quadrant II.