3 ## TRIGONOMETRY OF RIGHT TRIANGLESAll functions from one function PLANE TRIGONOMETRY is based on the fact of similar figures. (Topic 1.) We saw: Figures are similar if they are equiangular For triangles to be similar, however, it is sufficient that they be equiangular. (Theorem 15 of "Some Theorems of Plane Geometry.") From that it follows: Right triangles will be similar if an acute angle of one In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar. Therefore the sides that make the equal angles will be proportional. Whatever ratio CA has to AB, FD will have to DE. If CA were half of AB, for example, then FD would also be half of DE. A trigonometric Table is a table of ratios of sides. In the Table, each value of sin θ represents the ratio of the opposite side to the hypotenuse -- in If angle θ is 28°, say, then in sin 28° = .469 This means that in a right triangle having an acute angle of 28°, its opposite side is 469 thousandths of the hypotenuse, which is to say, a little less than half. It is in this sense that in a right triangle, the trigonometric ratios -- the sine, the cosine, and so on -- are "functions" of the acute angle. They depend only on the acute angle. Example. Indirect measurement. When we cannot measure things directly, we can use trigonometry. For example, to measure the height
From the Table, tan 37° = Therefore, on multiplying by 100,
All functions from one function If we know the value of any one trigonometric function, then -- with the aid of the Pythagorean theorem -- we can find the rest. Example 1. In a right triangle, sin θ = Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ.
To find the unknown side
Therefore,
We can now evaluate all six functions of θ:
Example 2. In a right triangle, sec θ = 4. Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ. To say that sec θ = 4, is to say that the hypotenuse is
To find the unknown side
Therefore,
We can now evaluate all six functions of θ:
Problem 1. In a right triangle, cos θ = . Sketch the triangle and evaluate sin θ. To see the answer, pass your mouse over the colored area. Problem 2. cot θ = . Sketch the triangle and evaluate csc θ. Two angles are called complements of one another if together they equal a right angle. Angles 1 and 2 are complements of one another. Each one completes the other to make a right angle. Thus the complement of 60° is 30°. This is the degree system of measurement in which a full circle, made up of four right angles at the center, is called 360°. (But see Topic 12: Radian Measure.) (Two angles are called supplements of one another if together they equal two right angles. In the system of degree measurement, 60° is the supplement of 120°. Their sum is 180°, which is two right angles.) Problem 3. Name the complement of each angle. a) 70° 20° b) 20° 70° c) 45° 45° d) θ 90° − θ The point about complements is that, in a right triangle, the two acute angles are complementary. For the three angles of the right triangle are together equal to (When we come to radian measure, we will see that 90° = ; therefore the complement of θ is − θ.) There are three pairs of cofunctions: The sine and the cosine The secant and the cosecant The tangent and the cotangent Here is the significance of a cofunction: A function of any angle is equal to the cofunction This means, for example, that sin 80° = cos 10°. The cofunction of the sine is the cosine. And 10° is the complement of 80°. Problem 4. Answer in terms of cofunctions. a) cos 5° = sin 85° b) tan 60° = cot 30° c) csc 12° = sec 78°
d) sin (90° − θ) = cos e) cot θ = tan (90° − In the figure:
Thus the sine of θ is equal to the cosine of its complement.
The secant of θ is equal to the cosecant of its complement.
The tangent of θ is equal to the cotangent of its complement. Next Topic: The 30°-60°-90° Triangle Please make a donation to keep TheMathPage online. Copyright © 2019 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |