The ratios of the lengths of the sides of a right triangle are called the trigonometric ratios. For convenience, we will name the three sides and three vertices of the right triangle as, a, b, and c for sides and the A, B, and C for the vertices as shown in the figure:
Sine (sin) Function of an acute angle of a right triangle is equal to the ratio of the length of the opposite leg to the length of the hypotenuse. Cosine (cos) Function of an acute angle of a right triangle is equal to the ratio of the length of the adjacent leg to the length of the hypotenuse. Tangent (tan) Function is equal to the ratio of the length of the opposite leg to the length of the adjacent leg. The reciprocal of the sine function is the Cosecant (csc) Function. The reciprocal of cosine and tangent are Secant (sec) and Cotangent (cot) function respectively. Right-Triangle-Based Definitions of Trigonometric Functions of angle A * sin A=ac
* cos A=bc
* tan A=ab
* csc A=ca
* sec A=cb
* cot A=ba
Right-Triangle-based definitions of Trigonometric Functions of angle B * sin B=bc
* cos B=ac
* tan B=ba
* csc B=cb
* sec B=ca
* cot B=ab
Lesson 2: Right Triangle
If this is the angle under consideration.
This side is called the opposite side because it is opposite the angle. This side is called adjacent side because it is near the angle. θ
A triangle in which one angle is a right angle is called a right triangle. In a right triangle, the three sides are given special names. a. The hypotenuse, which is the longest side, is the side opposite the 90° angle; and b. The remaining two sides are called the legs of the triangle. The two other sides are named in relation to another known angle (or unknown angle under consideration).
Lesson 3: Solution to Right Triangles
To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles. If we know the six trigonometric functions and the Pythagorean Theorem, we can find the unknown sides or angles of a given triangle. A right triangle can be solved if we have the following given: a. Two sides.
b. One side and an acute angle.
If one of the abovementioned are given, once one value is found, the second acute angle will be found using the fact that the two acute angles are complementary (A + B = 90°) and the remaining side by means of using a trigonometric function or the Pythagorean Theorem. The following are the suggested steps in solving a right triangle: 1. Draw the triangle and label the sides and the vertices. 2. Enter the given data.
3. Use the appropriate trigonometric functions to find for the unknown parts of the triangle.
b = 24
c = 25
To solve for a, use the Pythagorean Theorem;
a2 + b2 = c2
a2 = c2 - b2
a2 = c2-b2
a2 = 252-242
a2 = 625-576 = 49 = 7
a = 7.
Example 1: In a right triangle, cos A = 2425. Draw and label the triangle and solve for the remaining parts of the ∆.
For A, since the given is cos A = 2425;
cos A = 0.96
By means of using a calculator, press SHIFT then COS then = and °’”. We will get 16°.
To get the other acute angle use A + B = 90°. B = 90° - 16°= 74°.
Example 2: Find the missing parts of the triangle below:
The given are B = 56° and c = 50m.
Solve first for the measure of the remaining acute angle using: A = 90° - B = 90° - 56° = 34°.
Using the appropriate definition of the trigonometric function of angle B, we have; sin B=bc
sin 56°=b50 m
b = 50m sin 56° = 41 m
To solve for side a, we can use sin A=ac
sin 34°=a50 m
a=50 m (sin34°)
a= 28 m
or by means of the Pythagorean Theorem:
a=502-41.452 = 28 m
Lesson 4: Applications involving the...