It is necessary to calculate the sines of angles not only in a right triangle, but also in any other one. To do this, you need to draw the height of the triangle (a perpendicular to one of the sides, lowered from the opposite corner) and solve the problem as for a right triangle, using the height as one of the legs.
How to find the sine of an exterior angle of a triangle
First you need to understand what an external angle is. We have an arbitrary triangle ABC. If one of the sides, for example, AC, is extended beyond the angle BAC and a ray AO is drawn, then the new angle OAB will be external. This is the sine we will look for.
To solve the problem, we need to lower the perpendicular BH from angle ABC to side AC. This will be the height of the triangle. How we solve the problem will depend on what we know.
The simplest option is if the angle BAC is known. Then the problem can be solved extremely easily. Since the ray OS is a straight line, the angle OAS = 180°. This means that angles OAB and BAC are adjacent, and the sines of adjacent angles are equal in magnitude.
Let's consider another problem: in an arbitrary triangle ABC the side is known: AB=a and height ВН=h. We need to find the sine of the angle OAS. Since we now have a right triangle ABH, the sine of angle ABH will be equal to the ratio of the leg BH to the hypotenuse AB:
- sinBAH = BH/AB = h/a.
This is also simple. A more difficult task is if the height h and the sides AC=c, BC=b are known, and you need to find the sine of the angle OAB.
Using the Pythagorean theorem, we find the leg CH of the triangle BCH:
- BC² = BH² + CH² b² = h² + CH²,
- CH² = b² - h², CH = √(b² - h²).
From here you can find the segment AH of side AC:
- AH = AC - CH = c - √(b² - h²).
Now again we use the Pythagorean theorem to find the third side AB of triangle ABN:
- AB² = BH² + AH² = h² + (c - √(b² - h²))².
The sine of angle BAC is equal to the ratio of the height BN of the triangle to side AB:
- sinBAC = BH/AH = h/(c - √(b² - h²)).
Since angles OAB and BAC are adjacent, their sines are equal in magnitude.
Thus, by combining the Pythagorean theorem, the definition of sine and some other theorems (in particular, about adjacent angles), you can solve almost most problems about triangles, including finding the sine of an exterior angle. Sometimes additional constructions may be necessary: draw a height from the desired corner, extend the side of the corner beyond its limits, etc.
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By definition, any angle is made up of two divergent rays that emerge from a single common point - the vertex. If one of the rays is continued beyond the vertex, this continuation, together with the second ray, forms another angle - it is called adjacent. The adjacent angle at the vertex of any convex polygon is called external, since it lies outside the surface area limited by the sides of this figure.
Instructions
If you know the value of the sine of the internal angle (??) of a geometric figure, there is no need to calculate anything - the sine of the corresponding external angle (??) will have exactly the same value: sin(??) = sin(??). This is determined by the properties of the trigonometric function sin(??) = sin(180°-??). If it were necessary to find out, for example, the value of the cosine or tangent of an external angle, this value would need to be taken with the opposite sign.
There is a theorem that in a triangle, the sum of the values of any two internal angles is equal to the value of the external angle of the third vertex. Use it if the value of the internal angle corresponding to the external angle in question (??) is unknown, and the angles (?? and ??) at the other two vertices are given in the conditions. Find the sine of the sum of the known angles: sin(??) = sin(??+??).
A problem with the same initial conditions as in the previous step has a different solution. It follows from another theorem - about the sum of the interior angles of a triangle. Since this sum, according to the theorem, should be equal to 180°, the value of the unknown internal angle can be expressed through two known ones (?? and??) - it will be equal to 180°-??-??. This means you can use the formula from step one and replace the interior angle with this expression: sin(??) = sin(180°-??-??).
In a regular polygon, the value of the external angle at any vertex is equal to the value of the central angle, which means it can be calculated using the same formula as it. Therefore, if in the conditions of the problem the number of sides (n) of a polygon is given, when calculating the sine of any external angle (??), proceed from the fact that its value is equal to a full revolution divided by the number of sides. A complete revolution in radians is expressed by twice the number Pi, so the formula should look like this: sin(??) = sin(2*?/n). When calculating in degrees, replace double Pi by 360°: sin(??) = sin(360°/n).
In the section on the question, right triangle ABC is given, angle C is right. Find the sine of the external angle at vertex B, if AC = 3 and AB = 5 given by the author Anastasia Polupan the best answer is External angle of a triangle. Sine and cosine of external angle
Some USE problems require finding the sine, cosine or tangent of an exterior angle of a triangle. What is an exterior angle of a triangle?
Let's first remember what adjacent angles are. Here they are in the picture. Adjacent angles have one side in common, and the other two lie on the same straight line. The sum of adjacent angles is equal.
Adjacent angles
Let's take a triangle and extend one of its sides. An external vertex angle is an angle adjacent to a corner. If an angle is acute, then the angle adjacent to it is obtuse, and vice versa.
External angle of a triangle
Note that:
Remember these important relationships. Now we take them without evidence. In the section “Trigonometry”, in the topic “Trigonometric circle”, we will return to them.
It is easy to prove that the exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.
1. In a triangle, the angle is equal to, .Find the tangent of the exterior angle at the vertex.
External angle of a right triangle
Let be the external angle at the vertex.
Knowing this, we can find it using the formula
We get:
2. In a triangle, the angle is equal to, .Find the sine of the exterior angle at the vertex.
The problem is solved in four seconds. Since the sum of the angles and is equal, .Then the sine of the external angle at the vertex is also equal.
