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A short class with Sarah and her students. The other ways are under More.
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Today, let's explore trigonometry in right triangles! Can anyone tell me what a right triangle is?
It's a triangle with one angle that is 90 degrees!
Exactly! In a right triangle, we define key trigonometric ratios. Who can name one?
Sine!
Yes! The sine of angle A is the ratio of the length of the opposite side to the hypotenuse. It's expressed as sin(A) = opposite/hypotenuse. Let's remember that as O/H! Does everyone see the connection?
So, O/H means opposite over hypotenuse?
Correct! Now, what about cosine? Anyone?
Isn't it adjacent over hypotenuse?
Exactly! cos(A) = adjacent/hypotenuse. Great job! Let’s do a quick quiz. If we have a hypotenuse of 10 and an opposite side length of 6, what is sin(A)?
That would be 6/10 or 0.6!
Well done! You've grasped the first ratio. Remember, sin²A + cos²A = 1 is another important identity. Let's move on to tangent next!
Now that we know sine and cosine, let's talk about tangent. Who can tell me what tangent is?
Isn't it opposite over adjacent?
Right, that's tan(A) = opposite/adjacent. So if we had a right triangle where the opposite side is 4 and the adjacent side is 3, what's tan(A)?
It should be 4/3!
That's correct! Also, remember that we can relate tangent through sine and cosine, like tan(A) = sin(A)/cos(A). This ties everything together! Can someone remind me of the identity we discussed?
sin²A + cos²A = 1!
Exactly! You all are doing great. Let's summarize what we've learned so far: the definitions of sine, cosine, and tangent, along with their identities. Would anyone like to see an example?
Yes, please!
Let's apply what we've learned. Consider triangle ABC where AB = 13 and BC = 12. We need to find AC and sin(A). How would we start?
We can use the Pythagorean theorem to find AC!
Correct! Can someone tell me the theorem's formula?
It's a² + b² = c²!
Great! So, we can set up the equation: AC² = 13² - 12². What is that?
That would be 5!
Excellent! Now we found AC. What’s the sine of angle A?
It's 5/13!
Well done! Always remember to visualize how these ratios relate in triangles. We’re mastering trigonometry together!
Sine and Cosine, so divine; Opposite and Hypotenuse, across the line.
Imagine a sailor navigating a triangle, using sine to measure how far offshore he is while keeping his eye on the shoreline—sine helps him navigate.
SOHCAHTOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
CHOAS for remembering sides in triangle: C (Cos), H (Hypo), O (Opposite), A (Adjacent), S (Sin), T (Tan).
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What is the formula for sine?
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What is the formula for cosine?
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What is the formula for tangent?
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What is the Pythagorean Identity?
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The ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
The ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
The ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
The side opposite the right angle in a right triangle, and the longest side overall.
An equation that is true for all values of the variable, such as trigonometric identities.
This section covers trigonometric ratios for right triangles, including sine, cosine, and tangent.
In this section, we delve into the fundamental trigonometric ratios defined for right triangles, specifically sine, cosine, and tangent. We also explore basic identities that relate these functions and demonstrate their application through worked examples.
In right triangles, trigonometric ratios are defined based on the relationships between the angles and sides. For a right triangle labeled as ΔABC with angle C being the right angle, the following ratios are established:
Sine (sin): Defined as the ratio of the length of the side opposite the angle (A) to the hypotenuse. Thus,
sin(A) = opposite/hypotenuse = a/c
Cosine (cos): Defined as the ratio of the length of the adjacent side to the hypotenuse:
cos(A) = adjacent/hypotenuse = b/c
Tangent (tan): Defined as the ratio of the length of the opposite side to the adjacent side:
tan(A) = opposite/adjacent = a/b
Additionally, two core identities emerge from these definitions:
sin²A + cos²A = 1tan(A) = sin(A) / cos(A)These relationships enable the calculation of unknown side lengths and angles within right triangles.
Consider a right triangle ΔABC, where AB (hypotenuse) equals 13 and BC (one leg) equals 12. To find AC and sin(A):
Using the Pythagorean theorem, we calculate AC:
AC = √(AB² - BC²) = √(13² - 12²) = √(169 - 144) = √25 = 5
Now, using the sine function:
sin(A) = AC / AB = 5 / 13 ≈ 0.3846
This section underscores the importance of trigonometric functions for solving problems related to right triangles by leveraging their defined relationships.