Tangent of an Angle

1. Tangent of An Angle

The tangent of an angle can be said to be the ratio of the specific side opposite the angle above or over the side adjacent to the particular angle. Through the utilization of such information, one actually knows the length of the Side ST can be said to be 8 units as well as the length of the Side TU can be said to be 15 units.

2. Properties of the Tangent Function

The tangent function can be said to be expressed in the form of tan x = sin x/cos x as well as tan x = Perpendicular/Base. The slope of any straight line can be said to be the tangent of the particular angle that is made by the line along with the positive x-axis.

The values in relation to the tangent function at the specific angles can be said to be the following: -

The tan 0 = 0.

The tan π/6 = 1/√3.

The tan π/4 = 1.

The tan π/3 = √3.

The tan π/2 = Not defined.

3. What is the Law of Tangents?

In the particular trigonometry, the law of tangents can be said to be a statement regarding the relationship between the tangents of the 2 angles of any triangle as well as the lengths of the opposing or conflicting sides. As mentioned earlier, a, b, as well as c can be said to be the lengths of the 3 sides of the particular triangle, as well as α, β, and γ can be said to be the angles opposite those 3 respective sides. If you need any help with your Trigonometry Homework then you take the Trigonometry Homework Help from pro experts.

The trigonometric law of the tangents can be said to be the relation between the 2 sides of any plane triangle as well as the tangents of the specific sum as well as the difference of the angles opposite the particular sides. In any plane triangle ABC, if a, b, and care the sides opposite angles A, B, and C, correspondingly, then

The formula is particularly useful in the making of the calculations through the utilization of the logarithms.

4. Tangents for Special Common Angles

Sine and cosine cannot be said to be the only trigonometric functions utilized in trigonometry. Several others have been utilized throughout the ages, things such as the haversines and the spreads. The most valuable of these can be said to be the tangent. In the terms of the unit circle diagram, the tangent can be said to be the length of the vertical line ED tangent in relation to the circle from the specific point of the tangency E to the point D where that specific tangent line actually cuts the ray AD creating the angle.

Tangent in the terms of the sine and the cosine

Since the 2 triangles, ADE and ABC can be said to be similar, one shall have

ED / AE = CB / AC.

But ED = tan A, AE = 1, CB = sin A, and AC = cos AB. Therefore, one shall have derived the fundamental identity

Tangents and the right triangles

Just as the sine and the cosine can be found as the ratios of the sides of the right triangles, so can the specific tangent.

 

 

Slopes of the lines

A specific reason that the tangents can be said to be so significant is that they provide the slopes of the straight lines. One should consider the straight line that has been drawn in the x-y coordinate plane.

Point B can be said to be where the line actually cuts the y-axis. One might let the coordinates of the B be (0,b) so that b, known as the y-intercept, indicates how distant above the x-axis the B actually lies. (This notation actually conflicts with the labeling of the sides of any triangle a, b, as well as c, so one should not label the sides right presently.)

One shall be able to see that the point one unit to the right of the specific origin is labeled 1, as well as its coordinates, of course, can be said to be (1,0). Let C be the point where that specific vertical line actually cuts the specific horizontal line through B. Then C shall have the coordinates (1,b).

Point A is actually where the vertical line above one cuts the specific original line. Let m signify the distance that A is actually above C. Then A shall have coordinates (1,b+m). This value m is known as the slope of the specific line. If one moves right 1 unit wherever along the line, then one shall be able to move up the m units.

Presently, one should consider the angle CBA. One should call it the angle of the slope. The tangent can be said to be CA/BC = m/1 = m. Hence, the slope can be said to be the tangent of the angle of the slope.

5. Tangent Tables Chart 0° to 90°

6. Graphical Representation of the Tangent Function

In order to graph the specific tangent function, one actually marks the angle along the specific horizontal x axis, as well as for each angle, one actually puts the tangent of that angle upon the vertical y-axis. As perceived above, the result can be said to be rather the jagged curve that goes to the positive infinity in a single direction as well as negative infinity in the other.

7. Relationship Between Tangent and Other Trigonometric Functions

The 6 trig functions can be assembled in pairs as the reciprocals. Therefore, the sine value of any specific angle can always say to be the reciprocal of the specific cosecant value, as well as vice versa. For instance, if begin{align*}sin theta = frac{1}{2}end{align*}, then begin{align*}csc theta = frac{2}{1} = 2end{align*}.