Matching ratios to trig functions | Trigonometry

Understanding Trigonometric Ratios through Diagrams

In this article, we’ll dive into how trigonometric functions like sine, cosine, and tangent relate to angles using diagrams. We’ll focus on angle MKJ, which we’ll call angle theta (θ), and see how these functions work with this angle.

Trigonometric Functions and Their Definitions

First, let’s remember the basics of trigonometric functions using a right triangle. The acronym “SOH CAH TOA” helps us remember:

• Sine (sin): Opposite side over Hypotenuse
• Cosine (cos): Adjacent side over Hypotenuse
• Tangent (tan): Opposite side over Adjacent side

We can also use the unit circle to understand these functions:

• The cosine of an angle is the X-coordinate where the angle’s terminal side meets the unit circle.
• The sine of the angle is the Y-coordinate at that intersection.

Analyzing the Ratios

Cosine of Angle MKJ

Let’s start with the ratio ( frac{X}{1} ):

• Here, ( X ) is the length of the adjacent side to angle theta, and ( 1 ) is the hypotenuse in the unit circle.
• According to SOH CAH TOA, this ratio equals the cosine of angle MKJ:

[frac{X}{1} = cos(theta) = cos( ext{angle MKJ})]

Sine of Angle MKJ

Next, we look at the ratio ( frac{Y}{1} ):

• Here, ( Y ) is the length of the opposite side to angle theta.
• This ratio corresponds to the sine function:

[frac{Y}{1} = sin(theta) = sin( ext{angle MKJ})]

Tangent of Angle MKJ

Now, let’s examine the ratio ( frac{X}{Y} ):

• This ratio is the adjacent side over the opposite side, which defines tangent:

[frac{X}{Y} = frac{ ext{adjacent}}{ ext{opposite}} = tan(theta) = tan( ext{angle MKJ})]

Exploring Other Ratios

J over K

Consider the ratio ( frac{J}{K} ):

• Here, ( J ) is the adjacent side, and ( K ) is the opposite side to angle MKJ.
• This ratio is the reciprocal of tangent:

[frac{J}{K} = frac{ ext{adjacent}}{ ext{opposite}} = frac{1}{tan(theta)}]

Since this isn’t one of our main choices, we can ignore it.

K over J

Now, let’s look at the ratio ( frac{K}{J} ):

• This is the opposite side over the adjacent side, which equals tangent:

[frac{K}{J} = tan(theta) = tan( ext{angle MKJ})]

M over J

Consider the ratio ( frac{M}{J} ):

• This is the hypotenuse over the adjacent side, which is the reciprocal of cosine:

[frac{M}{J} = frac{ ext{hypotenuse}}{ ext{adjacent}} = frac{1}{cos(theta)}]

This isn’t one of our main choices either.

J over M

Now, the ratio ( frac{J}{M} ):

• This is the adjacent side over the hypotenuse, which corresponds to cosine:

[frac{J}{M} = cos(theta) = cos( ext{angle MKJ})]

K over M

Finally, let’s examine the ratio ( frac{K}{M} ):

• This is the opposite side over the hypotenuse, which equals sine:

[frac{K}{M} = sin(theta) = sin( ext{angle MKJ})]

Conclusion

By exploring these ratios, we’ve connected trigonometric functions with their geometric representations. Understanding these relationships helps us grasp how sine, cosine, and tangent work with angles, making trigonometry easier to understand and apply.