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4.3 Right Triangle Trigonometry As derived from the Greek language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with the relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. With the development of calculus and physical sciences in the 17th century, a different perspective rose – one that viewed the classic trig relationships as functions with the set of real numbers as their domains. This text incorporates both those perspectives, but we will deal with the triangle view first. These are all related to right triangles. Relative to the angle 𝜽 (“theta”), the 3 sides of a triangle are the hypotenuse, the opposite side, and the adjacent side. Using these sides, you can form six ratios that define the 6 trig functions of the acute angle 𝜃. Sine (sin)= Cosecant (csc) = Cosine (cos) = Secant (sec) = Tangent (tan) = Cotangent (cot) = Ex 1: Use the triangle to find the values of the 6 trig functions of 𝜃. 3 4 There are some special angles that we frequently use in trig. We will now find the trig values of these angles. The 450 – 450 – 900 Triangle: 1 1 The 300 – 600 – 900 Triangle: 2 If we need to answer questions that involve these angles, we usually use the measures above, leaving them in square root form. Ex 2: Use the given values of sin 600 = a. Tan 600 √3 2 and cos 600 = ½ to find the following: b. Sin 300 Ex 3: A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.50. How tall is the tree? Ex 4: A person is 200 yards away from a river. Rather than walk directly to the river, the person walks 400 yards along a straight path to the river’s edge. Find the acute angle 𝜃 between this path and the river’s edge. Ex 5: Specifications for a loading dock ramp require a rise of 1 foot for each 3 feet of horizontal length. In the figure below, find the lengths of sides b and c and find the measure of 𝜃. C b 4 ft