Right Triangle Model and the Pythagorean Theorem Help (page 2)
Introduction to Right Triangle Model and the Pythagorean Theorem
We have just defined the six circular functions – sine, cosine, tangent, cosecant, secant, and cotangent – in terms of points on a circle. There is another way to define these functions: the right-triangle model .
Triangle and Angle Notation
In geometry, it is customary to denote triangles by writing an uppercase Greek letter delta (Δ) followed by the names of the three points representing the corners, or vertices , of the triangle. For example, if P, Q , and R are the names of three points, then Δ PQR is the triangle formed by connecting these points with straight line segments. We read this as “triangle PQR .”
Angles are denoted by writing the symbol ∠(which resembles an extremely italicized, uppercase English letter L without serifs) followed by the names of three points that uniquely determine the angle. This scheme lets us specify the extent and position of the angle, and also the rotational sense in which it is expressed. For example, if there are three points P, Q , and R , then ∠ PQR (read “angle PQR ”) has the same measure as ∠ RQP , but in the opposite direction. The middle point, Q , is the vertex of the angle.
The rotational sense in which an angle is measured can be significant in physics, astronomy, and engineering, and also when working in coordinate systems. In the Cartesian plane, remember that angles measured counterclockwise are considered positive, while angles measured clockwise are considered negative. If we have ∠ PQR that measures 30° around a circle in Cartesian coordinates, then ∠ RQP measures –30°, whose direction is equivalent to an angle of 330°. The cosines of these two angles happen to be the same, but the sines differ.
Ratios of Sides
Consider a right triangle defined by points P, Q , and R , as shown in Fig. 12-9. Suppose that ∠ QPR is a right angle, so Δ PQR is a right triangle . Let d be the length of line segment QP , e be the length of line segment PR , and f be the length of line segment QR . Let θ be ∠ PQR , the angle measured counterclockwise between line segments QP and QR . The six circular trigonometric functions can be defined as ratios between the lengths of the sides, as follows:
sin θ = e / f cos θ = d / f tan θ = e / d csc θ = f / e sec θ = f / d cot θ = d / e
The longest side of a right triangle is always opposite the 90° angle, and is called the hypotenuse . In Fig. 12-9, this is the side QR whose length is f The other two sides are called adjacent sides because they are both adjacent to the right angle.
Fig. 12-9. The right-triangle model for defining trigonometric functions. All right triangles obey the theorem of Pythagoras. Illustration for Problems 12-5 and 12-6.
Sum of Angle Measures and Range of Angles
Sum of Angle Measures
The following fact can be useful in deducing the measures of angles in trigonometric calculations. It’s a simple theorem in geometry that you should remember. In any triangle, the sum of the measures of the interior angles is 180° (π rad). This is true whether it is a right triangle or not, as long as all the angles are measured in the plane defined by the three vertices of the triangle.
Range of Angles
In the right-triangle model, the values of the circular functions are defined only for angles between, but not including, 0° and 90° (0 rad and π/2 rad). All angles outside this range are better dealt with using the unit-circle model.
Using the right-triangle scheme, a trigonometric function is undefined whenever the denominator in its “side ratio” (according to the formulas above) is equal to zero. The length of the hypotenuse (side f ) is never zero, but if a right triangle is “squashed” or “squeezed” flat either horizontally or vertically, then the length of one of the adjacent sides ( d or e ) can become zero. Such objects aren’t triangles in the strict sense, because they have only two vertices rather than three.
The Right Triangle Model Practice Problems
Suppose there is a triangle whose sides are 3, 4, and 5 units, respectively. What is the sine of the angle θ opposite the side that measures 3 units? Express your answer to three decimal places.
If we are to use the right-triangle model to solve this problem, we must first be certain that a triangle with sides of 3, 4, and 5 units is a right triangle. Otherwise, the scheme won’t work. We can test for this by seeing if the Pythagorean theorem applies. If this triangle is a right triangle, then the side measuring 5 units is the hypotenuse, and we should find that 3 2 + 4 2 = 5 2 . Checking, we see that 3 2 = 9 and 4 2 = 16. Therefore, 3 2 + 4 2 = 9 + 16 = 25, which is equal to 5 2 . It’s a right triangle, indeed!
It helps to draw a picture here, after the fashion of Fig. 12-9. Put the angle θ , which we are analyzing, at lower left (corresponding to the vertex point Q ). Label the hypotenuse f = 5. Now we must figure out which of the other sides should be called d , and which should be called e . We want to find the sine of the angle opposite the side whose length is 3 units, and this angle, in Fig. 12-9, is opposite side PR , whose length is equal to e . So we set e = 3. That leaves us with no other choice for d than to set d = 4.
According to the formulas above, the sine of the angle in question is equal to e / f . In this case, that means sin θ = 3/5 = 0.600.
What are the values of the other five circular functions for the angle θ as defined in Problem 12-5? Express your answers to three decimal places.
Plug numbers into the formulas given above, representing the ratios of the lengths of sides in the right triangle:
cos θ = d / f = 4/5 = 0.800 tan θ = e / d = 3/4 = 0.750 csc θ = f / e = 5/3 = 1.667 sec θ = f / d = 5/4 = 1.250 cot θ = d / e = 4/3 = 1.333
Find practice problems and solutions for these concepts at: Trigonometry Practice Test.
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