Lesson 4.3 Our second approach to trigonometry is from a right triangle perspective…

Kuta WS Practice

Recall that a 30°-60°-90° triangle and a 45°-45°-90° triangle have special ratios: You can use special triangles to find the values of the following functions. Then check your answers on your unit circle. Refer to your definitions of the functions as needed. x 2 x 3

Trigonometry values for a given angle are always the same no matter how large the triangle is

With your partner … Discuss with your partner the patterns you notice in your answers from the previous slide. Jot at least 1 idea down. List the values that are the same and discuss why* these values might be the same. For example, notice that sin(30°) = 1 2 = cos(60°) Because 30 and 60 are complementary angles Cofunctions of complementary angles are equal *Because 30 and 60 are complementary angles

Where do the Pythagorean theorems come from?

Use the trig identities to transform one side of the equation into the other a) Cosθ sec θ = 1 b) (sec θ + tan θ)(sec θ - tan θ) =1

Make sure your calculator is in the correct mode.

Trig Applications EX 1 Find the length c of the skateboard ramp as shown in the figure. To help get started: In reference to the given angle, mark the opposite side, adjacent side, and hypotenuse. What ratio can we write using the given angle? sin(18.4°) = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 3) Fill in your values sin(18.4°) = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 4 𝑐 4) Rearrange to get the variable you are looking for by itself: c = 4 sin⁡(18.4°) = 12.67 feet

Definitions Angle of elevation – the angle formed by the line of sight and the horizontal plane (must be above the horizontal plane) Angle of Depression - the angle formed by the line of sight and the horizontal plane (must be below the horizontal plane) In the picture at the top right, an observer is standing at the top of a building and looking straight ahead at the birds (horizontal line). The observer must raise his eyes to see the airplane (slanting line). The angle formed between the two lines is called the angle of elevation.

Trig Applications EX 2 You are standing 50 feet from the base of a large tree, as shown in the figure. You estimate the angle of elevation (see next slide for definition) to the top of the tree as 71.5° How tall is the tree?

Trig Applications EX 3 You are 200 yards from a river. Rather than walk directly to the river, you walk 400 yards along a straight path to the river’s edge. Find the acute angle between your path and the river’s edge, as illustrated in the figure.

Worksheet Practice!

HW Poster presentations next day of hw

The Gallery: Trig Applications Solve the problem you get on your own. Make sure you draw an appropriate picture. Find your group members 3) Discuss with your group how to best represent your word problem visually. Use as much detail as possible in your final drawing. 4) Develop an equation using an appropriate trig ratio and solve for their missing side.

Presentations You are graded on: -Clarity -Organization -Accuracy

Example

Example