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A short class with Sarah and her students. The other ways are under More.
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Today we will explore the trigonometric ratios for 45°. Can anyone tell me about the properties of a triangle when one of the angles is 45°?
In a right triangle, if one angle is 45°, the other angle must also be 45°.
Exactly! This creates an isosceles right triangle where the two sides are equal. Now, if we denote those sides as 'a', how would we calculate the hypotenuse?
We use the Pythagorean theorem: AC² = AB² + BC², which means AC = a√2.
Correct! Now using this, can anyone derive sin 45°, cos 45°, and tan 45°?
Yes! Sin 45° = a / (a√2) = 1/√2, which is √2/2. Cos 45° will be the same, and tan 45° = 1.
Perfect! Remember the acronym 'SCC' for 'Sine, Cosine, Complementary' to help you recall that sin and cos are equal at 45°.
Let's summarize: For 45°, we have sin 45° = √2/2, cos 45° = √2/2, and tan 45° = 1.
Now, let’s discuss 30° and 60°. Who can tell me the relationship between these angles?
They are complementary angles, meaning they add up to 90°.
Exactly! We can derive their ratios using an equilateral triangle. If each side is 2a, what can we conclude?
We draw a height from one vertex to the opposite side, creating two 30-60-90 triangles.
Well done! In this scenario, how do we calculate sin 30°?
Sin 30° = opposite / hypotenuse, which is a / 2a = 1/2.
And for sin 60°?
Sin 60° = √3/2 because the opposite side is a√3 and the hypotenuse is 2a.
Great! The key angles to remember: sin 30° = 1/2, sin 60° = √3/2. Now let’s summarize.
Let’s finalize by discussing sin and cos for 0° and 90°. What can we observe as we approach these angles?
As the angle approaches 0°, the opposite side length approaches 0, which makes sin 0° = 0.
Exactly! And what about cos 0°?
Cos 0° = 1 since the hypotenuse remains unchanged.
Right! Now, as we move towards 90°, what happens to sin and cos?
Sin 90° becomes 1, and cos 90° becomes 0 as the triangle squashes.
Well summarized! For 0°: sin 0° = 0, cos 0° = 1; and for 90°: sin 90° = 1, cos 90° = 0. Let’s recap that!
At 30° and 60° in trigonometry's dance, sin and cos take their chance.
Imagine a triangle where an angle shines bright at 45°. The pairs of sides represent equal delight!
Acronym SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
For 45° remember SCC: Sine, Cosine, Correspond equally.
Term
What is sin 45°?
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Term
What is cos 30°?
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What is tan 90°?
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Relational ratios of the sides of a right triangle, defining the sine, cosine, tangent, cosecant, secant, and cotangent functions.
Ratio of the length of the opposite side to the hypotenuse in a right triangle.
Ratio of the length of the adjacent side to the hypotenuse in a right triangle.
Ratio of the length of the opposite side to the adjacent side in a right triangle.
Reciprocal of sine, or 1/sin.
Reciprocal of cosine, or 1/cos.
Reciprocal of tangent, or 1/tan.
This section explores the values of trigonometric ratios for the angles of 0°, 30°, 45°, 60°, and 90° using geometric constructions.
Students will learn to calculate trigonometric ratios for specific angles like 0°, 30°, 45°, 60°, and 90°. This section derives these values through geometric constructions and the properties of right triangles, emphasizing the significance in trigonometry.
In this section, we delve into the trigonometric ratios of key angles of 0°, 30°, 45°, 60°, and 90° through the application of right triangle properties and geometric constructions. We begin with calculating ratios for 45°, highlighting that in a right triangle where both angles are 45°, the legs are equal in length. By utilizing the Pythagorean theorem, we formulate the ratios, demonstrating that:
Next, we explore the ratios for angles 30° and 60° using an equilateral triangle. By drawing a perpendicular from one vertex, we create two right triangles, showing that:
Finally, we address the limits of trigonometric functions as angles approach 0° and 90°, defining values for:
The section concludes by summarizing the values of all calculated trigonometric ratios in a convenient table for reference.