Download 4.3 Right Triangle Trigonometry In the unit circle, we have the

4.3 Right Triangle Trigonometry
In the unit circle, we have the following values for the six trig functions:
However, if we expand outside the unit circle, we get some new values for cos θ and sin θ. Consider the following:
The following ratios hold for the six trig functions in any right triangle.
Use the Pythagorean Theorem to find the length of the missing side of the right triangle below. Then find the value of each of the six trigonometric functions of θ.
4
θ
9
1
Use the Pythagorean Theorem to find the length of the missing side of the right triangle below. Then find the value of each of the six trigonometric functions of θ.
11
7
θ
Two positive angles are complements if their sum is 90o or π/2. In a right triangle, the two acute angles are always complements, or complementary angles. If two angles are complements, the sine of one equals the cosine of the other. Because of this relationship the sine and cosine are called cofunctions of each other. The name cosine is a shortened form of the phrase "complement's sine."
Examples:
Tangent and cotangent are also cofunctions, as are secant and cosecant.
Examples: Find a cofunction with the same value as the given expression.
1) sin 20o
2) cos π/7
3) tan 24o
4) csc 3π/11
5) sec 50o
6) cot 5π/32
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Find the measure of the side of the right triangle below whose length is designated by a lowercase letter. Round answers to the nearest whole number.
220 cm
a
63o
34o
b
5 cm
If we are given the trig function value of an angle and need to know what the angle is, we can use the inverse trig functions. For example, to find θ, given sin θ = 2/5, we find sin­1(2/5).
Use a calculator to find the value of the acute angle θ to the nearest degree. (Calculator should be in degree mode.)
1) sin θ = 0.3578
2) cos θ = 0.9125
Use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. (Calculator should be in radian mode.)
3) tan θ = 1.357
4) sin θ = 0.683
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Many applications of right triangle trigonometry involve angles of elevation or angles of depression. An angle of elevation is measured from the horizontal up, and an angle of depression is measured from the horizontal down, as shown in the figure below, found on p. 495 of your book.
Example:
From a certain point on the ground, the sun is at an angle of elevation of 50o. If a tree between the point and the sun casts a shadow that is 28 feet long, determine the height of the tree.
A tower that is 100 feet tall casts a shadow 200 feet long. Find the angle of elevation to the sun, to the nearest degree.
A telephone pole is 80 feet tall. A guy wire 120 feet long is attached from the ground to the top of the pole. Find the angle between the wire and the pole to the nearest degree.
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