Topic 2 The Sine Law Unit 3 Topic 2

Before We Start

Explore We are going to develop the Sine Law using the oblique triangle below: ▫ 1) Draw in the height of the triangle and label it h. ▫ 2) Using the two right triangles formed, write a trigonometric ratio for sin A and for sin C. ▫ 3) Using the ratios from step 2, isolate for h. ▫ 4) Since both equations are equal to h, equate them to eliminate h. ▫ 5) Divide both sides of the equation by ac. AC B c b a

You should get… When using the steps on the previous slide, you should have gotten the following: AC B c b a

Information The Sine Law is a relationship between the sides and angles in an oblique triangle. C a b c AB

Example 1 Determining the length In  DEF, calculate the length of d to the nearest tenth. a)b) Try this on your own first!!!!

Example 1a: Solution Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’ Next we identify the side we are solving for and the given angle across from it. This forms our ‘incomplete set.’ d Set up the equation. Cross multiply. Divide.

Example 1b: Solution Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’ Next we identify the side we are solving for and the given angle across from it. This forms our ‘incomplete set.’ d Set up the equation. Cross multiply. Divide. In this question, we don’t have the angle across from the side we are solving for. We can find it using the triangle sum theory. 180 ° -27 ° -38 ° = 115° 115°

Example 2 Determining the angle Try this on your own first!!!!

Example 2a: Solution Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’ Next we identify the angle we are solving for and the given side across from it. This forms our ‘incomplete set.’ Set up the equation. Cross multiply. Divide. Use the inverse of sine to solve for the angle.

Example 2b: Solution Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’ Next we identify the angle we are solving for and the given side across from it. This forms our ‘incomplete set.’ Set up the equation. Cross multiply. Divide. Use the inverse of sine to solve for the angle.

Example 3 Determining the lengths and angle Try this on your own first!!!!

Example 3: Solution 110° Side ACSide BC Identify the complete set and the incomplete step to set up the equation for each. Cross multiply. Divide.

Example 4 Determining the length given two triangles Calculate the height, h, of the cliff given the following diagram, to the nearest metre. Try this on your own first!!!! In order to solve for h in the right-angled triangle, I first need the side common to both triangles. I need to use sine law to solve for the red side. I need to use the third (missing) angle to come up with my complete ratio. 66° a

Example 4: Solution (continued) h 31  65  185 m 49  Now we have the known side that we need in order to find h. We can use the first triangle, and label it according to the reference angle 31 ⁰ m Since we have the opposite side and the adjacent side, we can use the tan ratio. 66  opposite adjacent

Need to Know: A triangle that does not contain a right angle is called an oblique triangle. When solving for unknown values in an oblique triangle, the Pythagorean Theorem and SOH CAH TOA cannot be used.

Need to Know: If given a side length and the angle opposite to it, the Sine Law can be used to find the missing side length or angle. When finding a missing angle, use the inverse of sine (sin -1 ). You’re ready! Try the homework from this section.