Use the information from diagram to complete the table of trigonometric ratios. Place the ratio into the appropriate cell.
As the measure of the angle increases, the sine ratio increases, and the cosine ratio decreases. Furthermore, some values are repeated. For example, the sine of is the same as the cosine of and the cosine of is the same as the sine of This relationship leads to a rule.
It follows that the sum of the measures of and is Therefore, they are complementary angles. The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled and
Using the definitions of sine and cosine, the following equations can be obtained. By the Transitive Property of Equality, it can be said that and that
This is true for all pairs of complementary angles.
Write the given expression in terms of sine. Write your answer without the degree symbol.
Determine the value of that makes the equation true.
Magdalena is curious to determine if a right triangle exists where the sine and cosine of one of its acute angles have the same value. To do so, she lets be the measure of the angle and writes the following equation. She is considering drawing a diagram and using the definitions of sine and cosine.
The tangent of an acute angle is equal to the cotangent of its complementary angle. Similarly, the cotangent of an acute angle is equal to the tangent of its complementary angle. Therefore, the following statements hold true.
Consider a right triangle with side lengths and
By using their definitions, the tangent and cotangent ratios can be written in terms of and Since the acute angles of a right triangle are complementary, and are complementary angles. It can be seen that and This is true for all pairs of complementary angles.
Consider the product of the first term and the last terms of the expression.
In physics, the phenomenon known as refraction of light is described as the change in a light's direction as it passes from one medium to another. Due to refraction, objects in the water may appear to be closer to the water's surface than they actually are. The diagram shows Ignacio's eye, from above the surface of the ocean, viewing a whale that looks to have an apparent depth of feet below the surface of the ocean.
Begin by labeling points on the diagram.
As may have already been noticed, three of the trigonometric ratios start with the prefix
Consider an example trigonometric equation.
In this case, the prefix
co denotes that is the co-angle, or complementary angle, of The identities seen in this lesson are referred to as cofunction identities.