An Introduction to Trigonometry

Slideshow 44, MathematicsMr. Richard Sasaki, Room 307

Objectives• Learn and recall some components of the

right-angled triangle• Understand the meaning of sine, cosine and

tangent in terms of triangles• Understand the graphs

The Right-Angled TriangleTrigonometry is like Pythagoras but includes angles. When we have a specified angle, the vocabulary is different.

𝜃Angle

(Theta)

HypotenuseOpposite

Adjacent

Simple trigonometry involves 2 edges and an angle. If one thing is missing, how do we find it?

Special Case TrianglesWe saw two basic cases, the 30-60-90 triangle and the 45-45-90 triangle.

30𝑜1 2

√345𝑜

1 √2

1We 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 TangentSine, cosine and tangent are the relationships between edge lengths and angles.In calculation, sine, cosine and tangent are shown as , and respectively. We still usually refer to them by their full names though.Each refer to two of the edges.

𝜃

Hypotenuse

Opp

osite

Adjacent

SineCosineTangentS

O HC

A HT

O AYou will need to remember these links.

𝜃

Sine, Cosine and TangentIn fact, sine, cosine and tangent are functions on angles which equates to the ratio of the corresponding two edges.

Hypotenuse

Opp

osite

Adjacent

𝑠𝑖𝑛 (𝜃 )=¿𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑐𝑜𝑠 (𝜃 )=¿𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑡𝑎𝑛 (𝜃 )=¿𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

hypotenuse

adjacent

opposite

0 90 0101 0∞

As , neither are the hypotenuse so there is no restriction between their sizes.

12 √3

2

√33√2

2√221

Trigonometric FunctionsUsing the same methods, we can calculate trigonometric values for .

30𝑜60𝑜1 2

√3

𝑠𝑖𝑛 (60 )=¿𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=¿√32

𝑐𝑜𝑠 (60 )=¿𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=¿12

𝑡𝑎𝑛 (60 )=¿𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡=¿√3With a little more imagination, we can do the

same for and .

BoundariesWe do not have time to explore further but after looking at many values inserted in the trigonometric functions, we would have the following boundaries:

≤ 𝑠𝑖𝑛 (𝜃 )≤−1 1≤𝑐𝑜𝑠 (𝜃 )≤−1 1¿ 𝑡𝑎𝑛 (𝜃 )<¿−∞ ∞

This means the – axes corresponding to these graphs must obey these rules. What do these graphs look like?

We know the graph must satisfy .We saw that , and . This isn’t enough data to draw it, but it looks like this:

12

√22

√32

1

−1

𝑓 (𝑥)

180 360 540 720 90090

Note: We are using degrees, not radians.

Again, the graph satisfies .We saw that , and . It’s the opposite, right? Basically it’s like but shifted back :

√32

√22

12

1

−1

𝑓 (𝑥)

180 360 540 720 90090

This graph has no boundaries about .What is happening? , and .

√33 1

√3

4

−4

𝑓 (𝑥)

180 360 540 720 90090

The sizes are increasing.There is a cycle however…like this:

Note: tends to infinity.