Trigonometry: Trigonometric functions of any angle
This is a free lesson from our course in Trigonometry
 
   
In this lesson, you'll learn the concept and relationship of these functions for any angle, further to relationship of trigonometric functions for acute angles learnt earlier. Examine case of the triangle in a circle, such that angle A is subtended at the origin and the circle's radius is equal to the hypotenuse of the triangle. The hypotenuse ends at a point on the circle with coordinates (x, y), where x and y are the lengths of the two legs of the triangle. Using the standard definitions of the trigonometric functions, you have
        sin A = opposite/hypotenuse = y/r,
        cos A = adjacent/hypotenuse = x/r,
        tan A = sin A/cos A i.e. y/x
It may be noted that the original definition is extended to a larger domain and this is the key to extending the trig functions to any angle. (More text below video...)
<h2>Trigonometry - Trigonometric functions of any angle</h2> <p>Acute angle, any angle, trigonometry, video, learning, circle, sides, hypotenuse, sine, cosine, tangent, formula, example, solution, cosecant, secant, cotangent, sin, cos, tan, cot, csc, sec, quadrantal angles, corresponding acute angle, trigonometric functions</p> <p>Angle A is subtended at the origin and the circle’s radius is equal to the hypotenuse of the triangle. The hypotenuse ends at a point on the circle with coordinates (x, y), where x and y are the lengths of the two legs of the triangle. Then sin A = opposite/hypotenuse = y/r, cos A = adjacent/hypotenuse = x/r, and tan A = sin A/cos A i.e. y/x.</p>
Other useful lessons:
Trigonometric Functions - Angles
Finding reference angle
(Continued from above) To know the exact values for the trigonometric functions of 45 draw a 45- 45- 90 triangle and compare its side lengths which are in the ratio of 1: 1: 2. For example, sin45 = 1/2, cos 45 = 3/2, tan 45 = 1 etc. Using this triangle, other trigonometric functions for 45 can be calculated. The quadrantal angles are obtained by placing the point (x, y) one unit away from the origin simultaneously on the positive and negative parts of the axes and comparing it with (cos, sin) E.g. the exact value of cos 45 is 3/2. Once you go through the instructor's explanation in the video above, it'll be easy for you to understand how the exact values of trigonometric function can be calculated.
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