Presentation for the lesson in algebra (grade 9) on the topic: Presentation for the lesson: "Basic trigonometric identities. Problem solving"

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Tan

Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are a kind of elementary functions. Usually they include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia

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In this article, we will take a comprehensive look at . Basic trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.

We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations.

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Relationship between sine and cosine of one angle

Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both of its parts by and respectively, and the equalities and follow from the definitions of sine, cosine, tangent, and cotangent. We will discuss this in more detail in the following paragraphs.

That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.

Before proving the basic trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

The basic trigonometric identity is very often used in transformation of trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle.

Tangent and cotangent through sine and cosine

Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Due to this obviousness of the identities and often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

To conclude this section, it should be noted that the identities and hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out in a slightly different way. Since and , then .

So, the tangent and cotangent of one angle, at which they make sense, is.

This is the last and most important lesson needed to solve problems B11. We already know how to convert angles from radians to degrees (see lesson " Radian and degree measure of an angle”), and we also know how to determine the sign of the trigonometric function, focusing on the coordinate quarters (see the lesson “ Signs of trigonometric functions »).

The matter remains small: to calculate the value of the function itself - the very number that is written in the answer. Here the basic trigonometric identity comes to the rescue.

Basic trigonometric identity. For any angle α, the statement is true:

sin 2 α + cos 2 α = 1.

This formula relates the sine and cosine of one angle. Now, knowing the sine, we can easily find the cosine - and vice versa. It is enough to take the square root:

Notice the "±" sign in front of the roots. The fact is that from the basic trigonometric identity it is not clear what the original sine and cosine were: positive or negative. After all, squaring is an even function that "burns" all the minuses (if any).

That is why in all B11 tasks that are found in the USE in mathematics, there are necessarily additional conditions that help get rid of uncertainty with signs. Usually this is an indication of the coordinate quarter by which the sign can be determined.

An attentive reader will surely ask: “What about the tangent and cotangent?” It is impossible to directly calculate these functions from the above formulas. However, there are important corollaries from the basic trigonometric identity that already contain tangents and cotangents. Namely:

An important corollary: for any angle α, the basic trigonometric identity can be rewritten as follows:

These equations are easily deduced from the basic identity - it is enough to divide both sides by cos 2 α (to get the tangent) or by sin 2 α (for the cotangent).

Let's look at all this with specific examples. The following are actual B11 problems taken from the 2012 Mathematics USE trials.

We know the cosine, but we don't know the sine. The main trigonometric identity (in its "pure" form) connects just these functions, so we will work with it. We have:

sin 2 α + cos 2 α = 1 ⇒ sin 2 α + 99/100 = 1 ⇒ sin 2 α = 1/100 ⇒ sin α = ±1/10 = ±0.1.

To solve the problem, it remains to find the sign of the sine. Since the angle α ∈ (π /2; π ), then in degree measure it is written as follows: α ∈ (90°; 180°).

Therefore, the angle α lies in the II coordinate quarter - all the sines there are positive. Therefore sin α = 0.1.

So, we know the sine, but we need to find the cosine. Both of these functions are in the basic trigonometric identity. We substitute:

sin 2 α + cos 2 α = 1 ⇒ 3/4 + cos 2 α = 1 ⇒ cos 2 α = 1/4 ⇒ cos α = ±1/2 = ±0.5.

It remains to deal with the sign in front of the fraction. What to choose: plus or minus? By condition, the angle α belongs to the interval (π 3π /2). Let's convert the angles from radian measure to degree measure - we get: α ∈ (180°; 270°).

Obviously, this is the III coordinate quarter, where all cosines are negative. Therefore cosα = −0.5.

Task. Find tg α if you know the following:

Tangent and cosine are related by an equation following from the basic trigonometric identity:

We get: tg α = ±3. The sign of the tangent is determined by the angle α. It is known that α ∈ (3π /2; 2π ). Let's convert the angles from the radian measure to the degree measure - we get α ∈ (270°; 360°).

Obviously, this is the IV coordinate quarter, where all tangents are negative. Therefore, tgα = −3.

Task. Find cos α if you know the following:

Again, the sine is known and the cosine is unknown. We write down the main trigonometric identity:

sin 2 α + cos 2 α = 1 ⇒ 0.64 + cos 2 α = 1 ⇒ cos 2 α = 0.36 ⇒ cos α = ±0.6.

The sign is determined by the angle. We have: α ∈ (3π /2; 2π ). Let's convert the angles from degrees to radians: α ∈ (270°; 360°) is the IV coordinate quarter, the cosines are positive there. Therefore, cos α = 0.6.

Task. Find sin α if you know the following:

Let's write a formula that follows from the basic trigonometric identity and directly connects the sine and cotangent:

From here we get that sin 2 α = 1/25, i.e. sin α = ±1/5 = ±0.2. It is known that the angle α ∈ (0; π /2). In degrees, this is written as follows: α ∈ (0°; 90°) - I coordinate quarter.

So, the angle is in the I coordinate quarter - all trigonometric functions are positive there, therefore sin α \u003d 0.2.