An Introduction to Trigonometry Slideshow 44, Mathematics Mr. Richard Sasaki, Room 307

Objectives Learn and recall some components of the right-angled triangle Understand the meaning of sine, cosine and tangent for triangles Understand the graphs 𝑦= 𝑠𝑖𝑛 𝑥 , 𝑦= 𝑐𝑜𝑠 𝑥 , 𝑦=𝑡𝑎𝑛⁡(𝑥)

The Right-Angled Triangle Trigonometry is like Pythagoras but includes angles. When we have a specified angle, the vocabulary is different. Opposite Hypotenuse Simple trigonometry involves 2 edges and an angle. If one thing is missing, how do we find it? 𝜃 Angle (Theta) Adjacent

Special Case Triangles We saw two basic cases, the 30-60-90 triangle and the 45-45-90 triangle. 2 2 1 1 30 𝑜 45 𝑜 3 1 We need to think about the relationship between the edge lengths and the angles. Any ideas? The relationship lies with the three main trigonometric functions, , and . sine cosine tangent

Sine, Cosine and Tangent Sine, cosine and tangent are the relationships between edge lengths and angles. In calculation, sine, cosine and tangent are shown as 𝑠𝑖𝑛, 𝑐𝑜𝑠 and 𝑡𝑎𝑛 respectively. We still usually refer to them by their full names though. Each refer to two of the edges. Cosine Tangent Sine S C T Hypotenuse Opposite O H A H O A 𝜃 You will need to remember these links. Adjacent

Sine, Cosine and Tangent In fact, sine, cosine and tangent are functions on angles which equates to the ratio of the corresponding two edges. 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑠𝑖𝑛 𝜃 = Hypotenuse 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Opposite 𝑐𝑜𝑠 𝜃 = 𝜃 Adjacent 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑎𝑛 𝜃 =

hypotenuse opposite adjacent 0 90 0 1 0 1 0 ∞ As 𝑡𝑎𝑛 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 , neither are the hypotenuse so there is no restriction between their sizes. 1 2 3 3 3 2 2 2 1 2 2

Trigonometric Functions Using the same methods, we can calculate trigonometric values for 60 𝑜 . 3 2 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 2 𝑠𝑖𝑛 60 = 60 𝑜 1 30 𝑜 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 1 2 3 𝑐𝑜𝑠 60 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = 𝑡𝑎𝑛 60 = 3 With a little more imagination, we can do the same for 0 𝑜 and 90 𝑜 .

Boundaries We do not have time to explore further but after looking at many values inserted in the trigonometric functions, we would have the following boundaries: −1 ≤ 𝑠𝑖𝑛 𝜃 ≤ 1 −1 ≤ 𝑐𝑜𝑠 𝜃 ≤ 1 −∞ < 𝑡𝑎𝑛 𝜃 < ∞ This means the 𝑦 – axes corresponding to these graphs must obey these rules. What do these graphs look like?

𝑦=𝑓 𝑥 =sin⁡(𝑥) We know the graph must satisfy −1≤𝑦≤1. 2 2 2 2 We saw that sin 30 = , sin 45 = and sin 60 = . This isn’t enough data to draw it, but it looks like this: 1 2 3 2 𝑓(𝑥) 1 90 180 360 540 720 900 −1 Note: We are using degrees, not radians.

𝑦=𝑓 𝑥 =cos⁡(𝑥) Again, the graph satisfies −1≤𝑦≤1. 3 2 2 2 3 2 2 2 We saw that cos 30 = , cos 45 = and cos 60 = . It’s the opposite, right? Basically it’s like sin⁡(𝑥) but shifted back 90 𝑜 : 1 2 𝑓(𝑥) 1 90 180 360 540 720 900 −1

𝑦=𝑓 𝑥 =𝑡𝑎𝑛(𝑥) This graph has no boundaries about 𝑦. 3 3 3 3 What is happening? tan 30 = , tan 45 = and tan 60 = . 1 3 The sizes are increasing. There is a cycle however…like this: 𝑓(𝑥) 4 90 180 360 540 720 900 −4 Note: tan⁡(90) tends to infinity.

360 sin⁡(360𝑘+𝑥) 360 cos⁡(360𝑘+𝑥) 180 tan⁡(180𝑘+𝑥) 𝜃=5 𝜃=70 𝜃=50 𝜃=75 𝜃=7 𝜃=75 1 2 1 2 1 1