Two sides and an angle (SSA) of a triangle are given....

Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$a=30, b=40, A=20^{\circ}$$

Key Concepts

Triangle Existence Conditions

When solving a triangle given SSA data, it is crucial to assess whether the provided measurements can form a valid triangle. This involves comparing the given side lengths with the altitude (or height) derived from the known angle. If the side opposite the given angle is less than the altitude, no triangle exists; if it equals the altitude, there is one right triangle; if it is greater than the altitude but less than the other known side, two distinct triangles are possible; and if it is equal to or greater than the other side, only one triangle can be formed.

SSA Ambiguous Case

The SSA ambiguous case refers to situations in triangle problems where two sides and a non-included angle are known. This configuration is called 'ambiguous' because, depending on the specific measures, it may result in no triangle, exactly one triangle, or two distinct triangles. This phenomenon arises due to the sine function's property that sin(?) = sin(180° - ?), which can allow for two different possible angles that satisfy the equation derived from the Law of Sines.

Law of Sines

The Law of Sines is a fundamental trigonometric relationship in any triangle that relates the lengths of its sides to the sines of its opposite angles. It is expressed as (side/sin(angle)) = (side/sin(angle)) = (side/sin(angle)). In problems involving the SSA condition, the Law of Sines is typically used to calculate unknown angles or sides, but care must be taken due to the potential for multiple solutions arising from the ambiguous case.

Transcript

00:01 In this problem, we have to find the altitude first to determine the number of triangles possible.

00:06 We know that h is equals to b into sine a by substituting the known values we get.

00:15 H is equals to 40 into sine 20 degree.

00:20 By simplifying it, we get altitude which is approximately equals to 13 .7 units.

00:27 Since a is greater than h and a is less than b, so two triangles are formed.

00:35 So here we have a is equal to 30 units, b is equal to 40 units and angle a is equal to 20 degree.

00:49 By using the law of signs we will find the value of angle b, that is a upon sign a is equals to b upon sign b.

01:01 By substituting the known values we get 30 upon sine 20 degree is equal to 40 upon sign b.

01:16 Here we have sine b is equal to 40 into sine 20 degree upon 30.

01:27 By simplifying it we get sine b which is approximately equals to 0 .46.

01:34 So there are two angles between 0 degree and 180 degree for 2.

01:39 Which sign b is approximately equals to 0 .46.

01:44 Here we have angle b1 which is approximately equal to 27 degrees and angle b2 which is approximately equal to 180 degree minus 27 degree which is equals to 153 degree.

02:04 We know that the sum of measures of the interior angles of a triangle is 180 degree so we will find angle c1 and c2 substitute 22 degree for angle a and 27 degree for b1, 153 degree for b2 in a plus b1 plus c1 is equal to 180 degree and a plus b2 plus c2 is equal to 180 degree...

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