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A short class with Sarah and her students. The other ways are under More.
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Today, we are going to learn about trigonometric ratios. Can anyone tell me what trigonometric ratios are?
Are they the relationships between the angles and sides of triangles?
Exactly! Trigonometric ratios specifically apply to right-angled triangles, where we can define three primary ratios: sine, cosine, and tangent. Let's start with the sine ratio. Can someone explain how we calculate it?
Sine is the opposite side divided by the hypotenuse.
Great! A simple way to remember that is 'SOH' from the mnemonic 'SOH-CAH-TOA', where SOH stands for Sine = Opposite/Hypotenuse. Now, what is the cosine ratio?
Cosine is the adjacent side divided by the hypotenuse, right?
That's right! And 'CAH' in our mnemonic helps us remember that. Finally, who can tell me what tangent is?
Tangent is the opposite divided by the adjacent!
Correct, and we remember that with 'TOA'. So, can anyone summarize these three ratios for me?
Sure! Sine is Opposite/Hypotenuse, Cosine is Adjacent/Hypotenuse, and Tangent is Opposite/Adjacent.
Excellent summary! To wrap up, remember these ratios as they are fundamental in solving right triangle problems.
Now that we understand the ratios, let's look at how we use them to find unknown sides or angles. What would you do if we have one angle and one side?
We could use the ratios to find the other sides!
Exactly! For instance, if we know the angle θ and the hypotenuse, to find the opposite side we multiply the hypotenuse by sin θ. Can anyone provide me an example using this method?
If θ is 30° and the hypotenuse is 10 cm, the opposite side would be 10 × sin 30°.
So that means the opposite side would be 5 cm!
Well done! Now, if we know two sides instead, how do we find an angle? Any thoughts?
We can use inverse functions like sin⁻¹, cos⁻¹, or tan⁻¹!
Correct! For example, if we know the opposite side and hypotenuse, we can find θ by using θ = sin⁻¹(Opposite/Hypotenuse). Lastly, how would we summarize what we learned today?
We learned how to use trigonometric ratios to find unknown sides or angles!
Perfect! That wraps up our session.
Ratio of opposite side to hypotenuse.
Ratio of adjacent side to hypotenuse.
Ratio of opposite side to adjacent side.
Cosecant, secant, and cotangent.
Allow calculation of angles from known sides.
Example 1: For a right triangle with angle θ = 30° and hypotenuse = 10 cm, the length of the opposite side is found using Opposite = Hypotenuse × sin θ, giving Opposite = 10 × sin 30° = 5 cm.
Example 2: For a right triangle with the adjacent side of 7 cm and θ = 45°, the hypotenuse can be calculated using Hypotenuse = Adjacent / cos 45°, resulting in Hypotenuse = 7 / (√2 / 2) = 7√2 ≈ 9.9 cm.
To find sine and cosine, oh what a task! Use the ratios, and don't hesitate to ask.
Once upon a time in a triangle, the sine, cosine, and tangent rushed to solve a case, but they needed to know their sides to see their faces.
SOH-CAH-TOA helps you remember: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
To remember the reciprocal ratios, think of CSSC: Cosecant, Secant, Cotangent.
Term
What is cosine?
Click to flip
Term
What does the tangent ratio represent?
Click to flip
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What is the formula for sine?
Click to flip
The longest side of a right-angled triangle, opposite the right angle.
The side opposite the angle of interest in a right-angled triangle.
The side next to the angle of interest, excluding the hypotenuse.
Ratio of the length of the opposite side to the length of the hypotenuse.
Ratio of the length of the adjacent side to the length of the hypotenuse.
Ratio of the length of the opposite side to the length of the adjacent side.
Ratios that provide the inverse relationships: cosecant, secant, and cotangent.
This section covers the fundamental trigonometric ratios defined for right-angled triangles and their applications in finding unknown sides and angles.
In this section, we delve into trigonometric ratios, specifically sine, cosine, and tangent, defined as the ratios of the lengths of sides in a right-angled triangle. Additionally, we explore their reciprocal ratios and learn how these concepts are utilized to calculate unknown components of triangles based on provided angles and sides.
Trigonometric ratios are critical for understanding the relationships between the angles and sides of right-angled triangles. In this section, we focus on the three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:
Each ratio is applicable for angles between 0° and 90° in right-angled triangles, providing tools for solving various problems involving triangles. Besides these primary ratios, the section also introduces their reciprocal ratios: cosecant (csc), secant (sec), and cotangent (cot).
The application of these ratios enables us to find unknown sides or angles through the use of inverse trigonometric functions. The Pythagorean theorem serves as the foundation for relating these ratios through identities like sin²θ + cos²θ = 1. Finally, the section is equipped with practical examples illustrating how to apply these ratios in solving right-angled triangle problems.