Trigonometric ratios are often used to solve right triangles, but they cannot be used to solve non-right, or oblique, triangles. For these triangles, the Law of Sines and the Law of Cosines are particularly useful because they can be used to solve any triangle, right or oblique. To solve a triangle using the Law of Sines or the Law of Cosines, three pieces of information must be known.
|Two angles and a side length||Law of Sines||The Triangle Angle Sum Theorem can be used to find the missing angle measure. Then the Law of Sines can be used to find the unknown side lengths.|
|Two side lengths and a non-included angle||Law of Sines||The Law of Sines can be used to solve for one of the unknown angle measures. Then the Triangle Angle Sum Theorem can be used to find the third angle measure. Finally, the Law of Sines can be applied one more time to find the unknown side length.|
|Three side lengths||Law of Cosines||The Law of Cosines can be used to find any of the unknown angle measures. Then, either the Law of Sines or the Law of Cosines can be used to find another missing angle measure. Finally, the Triangle Angle Sum Theorem can be used to find the third angle measure.|
|Two side lengths and their included angle||Law of Cosines||The Law of Cosines can be used to find the missing side length. Then, either the Law of Cosines or the Law of Sines can be used used to find a missing angle measure. Finally, the Triangle Angle Sum Theorem can be used to find the last angle measure.|
Zain is vacationing in Italy. They were in Pisa to see the famous Leaning Tower when a question came across their mind. What would the tower's height be if it was not a leaning tower? Zain's distance to the tower is meters and they can measure an angle of elevation of to the tower's top. Furthermore, the guidebook states that the tower's inclination is about
Remembering a Geometry lesson, they realize that the situation can be modeled using a non-right triangle. Help Zain calculate the height of the upright tower. Write the answer rounded to one decimal place.
The situation can be modeled using a non-right triangle.
As mentioned before, the Law of Sines and the Law of Cosines are valid for all types of triangles, including both right and non-right triangles. However, the definitions of the sine and cosine of an angle are given in terms of the ratios of a right triangle's sides.
Ignacio's grandparent wants to construct a fence for a quadrilateral piece of land. To find the perimeter of the land, he starts measuring its sides using an old trundle wheel. Unfortunately, after measuring just two sides, the trundle wheel breaks.
Ignacio wants to help his grandparent and, in an attempt to simplify the problem, he divides the land into two triangles. Then, by using a compass, he is able to measure the angles of these triangles.
Note that two side lengths and all the angle measurements are known in
Now, will be considered.
Finally, the perimeter will be calculated by adding all the side lengths. The perimeter of the piece of land is about meters.
Kriz is setting up for a free shots on an empty goal. When considering their distance to both goal posts, they realize that the Law of Cosines can be used to calculate the top measure of the angle in which they must kick the ball in order to score. They are practicing with a standard meter net. Help Kriz calculate this angle and score the goal! Write the answer rounded to one decimal place.
The distance from the tower at the left to the smartphone is about blocks. The distance from the tower at the right to the phone is about blocks. The towers are blocks apart. Therefore, the situation can be modeled using a triangle with three known sides lengths.
Because two angles and the included side are known, this problem can be approached by using the Law of Sines to find the distance between the helicopter and one of the radar stations. Once one of the missing side lengths is determined, trigonometric ratios can be used to find the altitude, or height, of the helicopter. The first thing to do is to find the unknown angle.
The Law of Sines states that for any triangle, the ratio of the sine of an angle to the length of its opposite side is constant. However, this is not just any constant. In fact it has an important geometrical interpretation.
Since two side lengths and the included angle are known, the Law of Cosines can be used to find the missing side length. Because the missing side is opposite to the known angle, finding the side length will allow to calculate the desired ratio.