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In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians. Therefore and represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle' and μέτρον (métron) 'measure') [1] is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths.
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (⁄4 turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side in the figure). The sides adjacent to the right angle are called legs ...
For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.. To define the sine and cosine of an acute angle , start with a right triangle that contains an angle of measure ; in the accompanying figure, angle in a right triangle is the angle of interest.
The shaded blue and green triangles, and the red-outlined triangle are all right-angled and similar, and all contain the angle . The hypotenuse B D ¯ {\displaystyle {\overline {BD}}} of the red-outlined triangle has length 2 sin θ {\displaystyle 2\sin \theta } , so its side D E ¯ {\displaystyle {\overline {DE}}} has length 2 sin 2 θ ...
Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles ...
Similar right triangles illustrating the tangent and secant trigonometric functions. Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ. The identities
A right triangle with sides relative to an angle at the point. Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that.
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