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In the diagram's right triangles, some angle measures and side lengths are shown. Find the missing angle measures and lengths.
Use the information from diagram to complete the table of trigonometric ratios. Place the ratio into the appropriate cell.Consider the following table of trigonometric ratios for some acute angles.
sinθ | cosθ | |
---|---|---|
θ=15∘ | 0.258819… | 0.965925… |
θ=30∘ | 0.5 | 0.866025… |
θ=45∘ | 0.707106… | 0.707106… |
θ=60∘ | 0.866025… | 0.5 |
θ=75∘ | 0.965925… | 0.258819… |
The sine of an acute angle is equal to the cosine of its complementary angle. Similarly, the cosine of an acute angle is equal to the sine of its complementary angle.
By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is 180∘. For a right triangle, since one angle measures 90∘, the other two angles are acute.
It follows that the sum of the measures of ∠A and ∠B is 90∘. Therefore, they are complementary angles.This is true for all pairs of complementary angles.
Write the given expression in terms of sine. Write your answer without the degree symbol.
The sine of an acute angle is equal to the cosine of its complement. Similarly, the cosine of an acute angle is equal to the sine of its complement.
Determine the value of x that makes the equation true.
Complementary angles add up to 90∘.
Substitute expressions
LHS−172x=RHS−172x
Commutative Property of Addition
Factor out x
Subtract fractions
LHS+171=RHS+171
Add fractions
b1⋅a=ba
LHS⋅17=RHS⋅17
Like the sine and cosine, the same relationship exists between the tangent and cotangent.
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 x, y, and z.
By using their definitions, the tangent and cotangent ratios can be written in terms of x and y.Use the tangent and cotangent relationship of complementary angles.
Consider the product of the first term and the last terms of the expression.
Since the complement of 89∘ is 1∘, by the tangent and cotangent relationship of complementary angles, it can be said that tan89∘ is equal to cot1∘. Recall that the cotangent is the reciprocal of the tangent. Therefore, the product of the first and the last terms is equal to 1. Similarly, the product of the second and second to last terms is equal to 1, and so on. The terms in this product can be grouped so that each pair has a product of 1. Since the given product has 89 terms, tan45∘ will stand alone.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 30 feet below the surface of the ocean.
Use the tangent and cotangent relationship of complementary angles.
Begin by labeling points on the diagram.
The relationship discussed throughout this lesson can be extended to include the secant and cosecant ratios.
As may have already been noticed, three of the trigonometric ratios start with the prefixco.
codenotes that β is the co-angle, or complementary angle, of α. The identities seen in this lesson are referred to as cofunction identities.
Cofunction Identities | |
---|---|
sinθ=cos(90∘−θ) | cosθ=sin(90∘−θ) |
tanθ=cot(90∘−θ) | cotθ=tan(90∘−θ) |
secθ= |