Trigonometry. Angles add to 180° презентация

to 180°

in which one of the angles is 90°
Here’s a right triangle:

We call the longest side the hypotenuse
We pick one of the other angles--not the right angle
We name the other two sides relative to that angle

Here’s the
right angle

hypotenuse

adjacent

opposite

A) This time we proceed directly to the calculation of the roots of the equation at once: x / 5 = ± arccos (1) + 2πk.
Then x / 5 = πk => x = 5πk The answer is: x = 5πk, where k is an integer.

b) tg (3x-π / 3) = √3
B) We write in the form: 3x-π / 3 = arctg (√3) + πk. We know that: arctg (√3) = π / 3 3x-π / 3 = π / 3 + πk => 3x = 2π / 3 + πk => x = 2π / 9 + πk / 3 The answer is: x = 2π / 9 + πk / 3, where k is an integer.

sides and add them, you get the square of the length of the hypotenuse
adj2 + opp2 = hyp2

32 + 42 = 52, or 9 + 16 = 25
hyp = sqrt(adj2 + opp2)
5 = sqrt(9 + 16)

formula
3-4-5 and its multiples (6-8-10, etc.) are the best known
5-12-13 and its multiples form another set

25 + 144 = 169

divide the lengths of the sides
Each of these six ratios has a name (and an abbreviation)
Three ratios are most used:
sine = sin = opp / hyp
cosine = cos = adj / hyp
tangent = tan = opp / adj
The other three ratios are redundant with these and can be ignored

The ratios depend on the shape of the triangle (the angles) but not on the size

all the roots on the interval [0; Π].
Decision: Let us solve our equation in general form: 4x = ± arccos (√2 ​​/ 2) + 2πk 4x = ± π / 4 + 2πk;
X = ± π / 16 + πk / 2;
Now let's see what roots get into our segment.

When k <0, the solution is also less than zero, we do not fall into our segment.
For k = 0, x = π / 16, we are in the given interval [0; Π].
For k = 1, x = π / 16 + π / 2 = 9π / 16, again we have got.
For k = 2, x = π / 16 + π = 17π / 16, and here we are no longer there, and therefore for large k we also will not fall.
The answer is: x = π / 16, x = 9π / 16

addition to the right angle) and the length of a side, you can compute all other angles and lengths of sides

If you know the angle marked in red (call it A) and you know the length of the adjacent side, then
tan A = opp / adj, so length of opposite side is given by opp = adj * tan A
cos A = adj / hyp, so length of hypotenuse is given by hyp = adj / cos A

is an integer.

б)tg(3x- π/3)= √3
The answer is: x = 2π / 9 + πk / 3, where k is an integer.

the roots of the equation at once:
x / 5 = ± arccos (1) + 2πk. Then x / 5 = πk => x = 5πk The answer is: x = 5πk, where k is an integer.

B) We write in the form: 3x-π / 3 = arctg (√3) + πk.
We know that:
arctg (√3) = π / 3 3x-π / 3 = π / 3 + πk => 3x = 2π / 3 + πk => x = 2π / 9 + πk / 3
The answer is:
x = 2π / 9 + πk / 3, where k is an integer

for doing all kinds of things with basic graphics
Here’s the part I’ve always found the hardest:
Memorizing the names of the ratios
sin = opp / hyp
cos = adj / hyp
tan = opp / adj