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In a right triangle with an angle of 30°, if the hypotenuse is 5 units long, what is the value of the adjacent side?
In a right triangle with an angle of 30°, if the hypotenuse is 5 units long, what is the value of the adjacent side?
For a right triangle with an angle of 45°, if the opposite side is 1 unit, what is the value of the hypotenuse?
For a right triangle with an angle of 45°, if the opposite side is 1 unit, what is the value of the hypotenuse?
When solving the equation (\cos x = 0.6), which angle between 0° and 360° satisfies this equation?
When solving the equation (\cos x = 0.6), which angle between 0° and 360° satisfies this equation?
For the equation (\tan x = \frac{1}{2}), which of the following angles between 0° and 360° satisfies this equation?
For the equation (\tan x = \frac{1}{2}), which of the following angles between 0° and 360° satisfies this equation?
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If (\sin 45^ ext{o} = 0.5), what is (\cos 45^ ext{o}) in a right triangle with a 45° angle?
If (\sin 45^ ext{o} = 0.5), what is (\cos 45^ ext{o}) in a right triangle with a 45° angle?
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In a right triangle with an angle of 60°, if the hypotenuse is 10 units long, what is the value of the opposite side?
In a right triangle with an angle of 60°, if the hypotenuse is 10 units long, what is the value of the opposite side?
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What is the definition of sine in trigonometry?
What is the definition of sine in trigonometry?
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Which trigonometric function is defined as the ratio of the adjacent side to the hypotenuse?
Which trigonometric function is defined as the ratio of the adjacent side to the hypotenuse?
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In a right triangle, which trigonometric function is the ratio of the side opposite the angle to the side adjacent to the angle?
In a right triangle, which trigonometric function is the ratio of the side opposite the angle to the side adjacent to the angle?
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What does cosθ equal to in trigonometry?
What does cosθ equal to in trigonometry?
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Which trigonometric function is defined as $\frac{opposite}{hypotenuse}$ in trigonometry?
Which trigonometric function is defined as $\frac{opposite}{hypotenuse}$ in trigonometry?
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What is $ an \theta$ in terms of sides in a right triangle?
What is $ an \theta$ in terms of sides in a right triangle?
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Study Notes
Math: Exploring Trigonometry and Exercise 2.12 for 11th Grade Students
Trigonometry, a branch of mathematics that deals with the relationships between the angles and sides of triangles, is a vital concept to understand as you progress through high school math. This article will focus on 11th grade trigonometry, specifically delving into the Exercise 2.12 from common textbooks. We'll begin by covering the basics of trigonometry and then dive into the exercise itself, providing explanations and examples.
Trigonometric Functions
In trigonometry, you'll encounter three primary functions: sine (sin), cosine (cos), and tangent (tan). These functions are defined in terms of the ratios of the sides of a right triangle.
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Sine (sin): The sine of an angle θ in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
[ \sin \theta = \frac{opposite}{hypotenuse} ]
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Cosine (cos): The cosine of an angle θ in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
[ \cos \theta = \frac{adjacent}{hypotenuse} ]
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Tangent (tan): The tangent of an angle θ in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
[ \tan \theta = \frac{opposite}{adjacent} ]
For example, in the right triangle shown above, θ is 30°, and the lengths are 3 (adjacent), 4 (opposite), and 5 (hypotenuse). In this case, (\sin 30^\circ = \frac{3}{5}), (\cos 30^\circ = \frac{4}{5}), and (\tan 30^\circ = \frac{3}{4}).
Exercise 2.12
Exercise 2.12 for 11th grade students typically involves manipulating trigonometric functions of complementary angles and solving trigonometric equations. Below we'll go through a few examples to help you understand the concepts and techniques involved.
Example 1:
Given (\sin 45^\circ = \frac{1}{2}), find the values of (\cos 45^\circ) and (\tan 45^\circ).
(\sin 45^\circ = \frac{1}{2} = \frac{opposite}{hypotenuse}). Since right triangles with 45° angles have two legs of equal length, the opposite and adjacent sides are equal. Therefore, the adjacent side is also 1, and:
[\cos 45^\circ = \frac{adjacent}{hypotenuse} = \frac{1}{2} \ \ \text{and} \ \ \tan 45^\circ = \frac{opposite}{adjacent} = \frac{1}{1} = 1]
Example 2:
Solve the equation (\cos x = 0.6) for (0^\circ \leq x \leq 360^\circ).
To solve this equation, we'll first find the angles for which (\cos x = 0.6). We know that the cosine function reaches its maximum and minimum values at specific angles.
[\cos 60^\circ = \cos 300^\circ = 0.5 > 0.6]
[\cos 300^\circ = \cos 360^\circ - 300^\circ = -0.5 < 0.6]
Now we look for angles between (0^\circ) and (360^\circ) where (\cos x = 0.6). We'll use the fact that (\cos (180^\circ - x) = -\cos x).
[\cos (180^\circ - x) = -\cos x = -0.6]
[\cos x = 0.6 = \cos (180^\circ - x)]
Now we can find the angles (x) by solving for (180^\circ - x):
[180^\circ - x = 120^\circ \Rightarrow x = 60^\circ]
Thus, the solution to the equation (\cos x = 0.6) is (x = 60^\circ).
Example 3:
Solve the equation (\tan x = \frac{1}{2}) for (0^\circ \leq x \leq 360^\circ).
To solve this equation, we'll first find the angles for which (\tan x = \frac{1}{2}). We know that the tangent function reaches its maximum and minimum values at specific angles:
[\tan 45^\circ = \tan 315^\circ = 1 > \frac{1}{2}]
[\tan 315^\circ = \tan (360^\circ - 45^\circ) = -1 < \frac{1}{2}]
Now we look for angles between (0^\circ) and (360^\circ) where (\tan x = \frac{1}{2}). We'll use the fact that (\tan (180^\circ - x) = -\tan x).
[\tan (180^\circ - x) = -\tan x = -\frac{1}{2}]
[\tan x = \frac{1}{2} = \tan (180^\circ - x)]
Now we can find the angles (x) by solving for (180^\circ - x):
[180^\circ - x = 135^\circ \Rightarrow x = 45^\circ]
Thus, the solution to the equation (\tan x = \frac{1}{2}) is (x = 45^\circ).
This exercise is just a small sample of what you’ll cover in 11th grade trigon
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Description
Explore trigonometry concepts and Exercise 2.12 for 11th grade students, focusing on trigonometric functions like sine, cosine, and tangent. Learn how to manipulate trigonometric functions of complementary angles and solve equations involving trigonometry. Examples provided for better understanding.
