Day 87 – Finding trigonometric ratios

Introduction We interact with slopes every day; Slope of increasing and decreasing prices of goods, slopes of terrain when going to school, visiting places, slopes in forex trading data, slopes in inclined planes among others. These slopes can be measured in terms of degrees. However, we have already introduced another way of trying to evaluate these slopes. These are the use of trigonometric ratios. In this lesson, we are going to find trigonometric ratios of acute angles.

Vocabulary Trigonometric ratios These are ratios of different combinations of two sides in a right angle triangle taken with respect to a given angle.

Trigonometric ratio We now understand the basic trigonometric ratios sine, cosine and tangent abbreviated as sin, cos and tan respectively. We also have slightly advanced ratios which are reciprocals of the above. These are cosecant, secant, and cotangent abbreviated as csc, sec and cot respectively. Given the angle 𝜃 is an acute angle in a triangle, and 𝑎, 𝑜, ℎ represents the adjacent side, the opposite side and the hypotenuse side respectively, then cos 𝜃= 𝑎 ℎ ; sin 𝜃= 𝑜 ℎ ; tan 𝜃= 𝑜 𝑎 csc 𝜃= 1 sin 𝜃 = ℎ 𝑜 ; sec 𝜃= 1 cos 𝜃 = ℎ 𝑎 ; cot 𝜃= 1 tan 𝜃 = 𝑎 𝑜

Finding trigonometric ratios We can find the trigonometric ratios of an angle given the sides or given the angle itself. (i). Given the sides Given the sides, we simply substitute into the formula for such ratio (ii). Given the angle Given the angle, we use the calculators or the trigonometric tables to read the angles. For the case of a calculator, we simply press the button for the required trigonometric ratio then key in the angle in degrees.

In this lesson, we will not discuss the use of trigonometric tables In this lesson, we will not discuss the use of trigonometric tables. Example 1 Find the sin 𝐴 , sin 𝐵 , cos 𝐴 , cos 𝐵 , tan 𝐴 𝑎𝑛𝑑 tan 𝐵 given the following diagram. 12 in 8 in A B C

Solution We first find the length of AB Solution We first find the length of AB. AB is the hypotenuse hence, by Pythagorean theorem, we have 𝐴𝐵= 𝐵 𝐶 2 +𝐶 𝐴 2 = 8 2 + 12 2 = 208 =14.42 𝑖𝑛 With respect to ∠𝐴, adjacent side = 12 in Opposite side = 8 in Thus, cos 𝐴 = 𝑎 ℎ = 12 14.42 =0.8322 sin 𝐴 = 𝑜 ℎ = 8 14.42 =0.5548 tan 𝐴 = 𝑜 𝑎 = 8 12 =0.6667

With respect to ∠𝐵, Adjacent side = 8 in Opposite side = 12 in Thus, cos 𝐵 = 𝑎 ℎ = 8 14.42 =0.5548 sin 𝐵 = 𝑜 ℎ = 12 14.42 =0.8322 tan 𝐵 = 𝑜 𝑎 = 12 8 =1.5

Example 2 Find the sin 𝑥 , cos 𝑥 , cot 𝑥 and csc 𝑥 given that 𝑥 =65° Example 2 Find the sin 𝑥 , cos 𝑥 , cot 𝑥 and csc 𝑥 given that 𝑥 =65°. Solution sin 65° We ensure the calculator is in degrees mode. We then press sin button, key in 65, the press = button. sin 65°=0.9063 Doing the same for cosine, we get cos 65°=0.4226 To get cot 65° , we find tan 65° then get its reciprocal. tan 65° =2.145

To get cot 65° , we find tan 65° then get its reciprocal. tan 65° =2 To get cot 65° , we find tan 65° then get its reciprocal. tan 65° =2.145 cot 65°= 1 tan 65° = 1 2.145 =0.4662 csc 65° = 1 cos 65° = 1 0.4226 =2.366

homework Using the following diagram, find sine and cosine of angle PML 8 in 6 in P M L

Answers to homework sin ∠𝑀=0.8 cos ∠𝑀=0.6

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