When is tangent 1

As with the sine and cosine functions, tangent is a periodic function. This means that its values repeat at regular intervals. If we look at any larger interval, we will see that the characteristics of the graph repeat. The graph of the tangent function is symmetric around the origin, and thus is an odd function.

Calculate values for the trigonometric functions that are the reciprocals of sine, cosine, and tangent. We have discussed three trigonometric functions: sine, cosine, and tangent. Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function.

Note that reciprocal functions differ from inverse functions. Inverse functions are a way of working backwards, or determining an angle given a trigonometric ratio; they involve working with the same ratios as the original function. It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle. It is easy to calculate secant with values in the unit circle.

Therefore, the secant function for that angle is. It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle. As with secant, cosecant can be calculated with values in the unit circle. Therefore, the cosecant function for the same angle is. It can be described as the ratio of the length of the adjacent side to the length of the hypotenuse in a triangle.

Cotangent can also be calculated with values in the unit circle. We now recognize six trigonometric functions that can be calculated using values in the unit circle.

Recall that we used values for the sine and cosine functions to calculate the tangent function. We will follow a similar process for the reciprocal functions, referencing the values in the unit circle for our calculations. In other words:. Describe the characteristics of the graphs of the inverse trigonometric functions, noting their domain and range restrictions.

Inverse trigonometric functions are used to find angles of a triangle if we are given the lengths of the sides. Inverse trigonometric functions can be used to determine what angle would yield a specific sine, cosine, or tangent value. Note that the domain of the inverse function is the range of the original function, and vice versa. However, the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test.

In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. These choices for the restricted domains are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible.

The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next, instead of being divided into pieces by an asymptote. We can define the inverse trigonometric functions as follows. Note the domain and range of each function. To find the domain and range of inverse trigonometric functions, we switch the domain and range of the original functions. Privacy Policy. Skip to main content. Search for:. Trigonometric Functions and the Unit Circle.

Learning Objectives Explain the definition of radians in terms of arc length of a unit circle and use this to convert between degrees and radians. Key Takeaways Key Points One radian is the measure of the central angle of a circle such that the length of the arc is equal to the radius of the circle. Key Terms arc : A continuous part of the circumference of a circle. The measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.

Learning Objectives Use right triangles drawn in the unit circle to define the trigonometric functions for any angle. The unit circle demonstrates the periodicity of trigonometric functions by showing that they result in a repeated set of values at regular intervals. Tan 1 degrees can also be expressed using the equivalent of the given angle 1 degrees in radians 0. The tangent function is positive in the 1st quadrant.

We can find the value of tan 1 degrees by:. Using trigonometry formulas, we can represent the tan 1 degrees as:. Tan 1 degrees is the value of tangent trigonometric function for an angle equal to 1 degrees. The exact value of tan 1 degrees can be given accurately up to 8 decimal places as 0. Learn Practice Download.

The tangent function is positive in the first and third quadrants. To find the second solution , add the reference angle from to find the solution in the fourth quadrant. To write as a fraction with a common denominator , multiply by. Write each expression with a common denominator of , by multiplying each by an appropriate factor of.

Multiply by. Combine the numerators over the common denominator. Simplify the numerator.