10.Sınıf Matematik 1.Dönem 1.Yazılı 1.TEMA Özet Tekrar | OYSS

Summary

The video reviews the main topics in a 10th-grade trigonometry and geometry unit. The instructor emphasizes recognizing which theorem or identity applies, drawing helpful right triangles, and using known triangle properties before starting calculations.

1. Trigonometric Ratios in Right Triangles

Trigonometric ratios are defined for an acute angle in a right triangle. Relative to angle (\alpha):

• Opposite side: the side across from (\alpha)
• Adjacent side: the leg beside (\alpha)
• Hypotenuse: the side opposite the right angle

The four ratios are:

[ \sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}} ]

[ \tan\alpha=\frac{\text{opposite}}{\text{adjacent}},\qquad \cot\alpha=\frac{\text{adjacent}}{\text{opposite}} ]

The instructor’s memory aid is to identify the sides relative to the angle first, then select the ratio. The hypotenuse appears in sine and cosine, but not in tangent or cotangent.

Useful strategies:

• If a problem does not already show a right triangle, draw one that includes the angle or angles needed.
• Use familiar side ratios such as (3!-!4!-!5) and (5!-!12!-!13).
• Treat a ratio such as (12/13) as a proportion: the corresponding sides could be (12k) and (13k). Use a given side length to find (k), then determine the remaining sides.
• For the same angle, tangent and cotangent are reciprocals.

2. Trigonometric Identities and Simplifying Expressions

Using the Pythagorean theorem in a right triangle gives the fundamental identity:

[ \sin^2x+\cos^2x=1. ]

The video demonstrates applying this identity by squaring a given sum or difference:

• If (\sin x+\cos x) is known, square it:

[ (\sin x+\cos x)^2 =\sin^2x+\cos^2x+2\sin x\cos x =1+2\sin x\cos x. ]

This can be used to find (\sin x\cos x).

• For a difference:

[ (\sin x-\cos x)^2 =1-2\sin x\cos x. ]

• If the product (\sin x\cos x) is given and the sum is required, let the sum be (A), square it, and substitute (\sin^2x+\cos^2x=1).

Other simplification techniques covered:

• Replace (1) with (\sin^2x+\cos^2x) when doing so creates cancellation.
• Factor expressions such as (1-\sin^2x) using the difference of squares:

[ 1-\sin^2x=(1-\sin x)(1+\sin x). ]

• Rewrite tangent and cotangent in terms of sine and cosine:

[ \tan x=\frac{\sin x}{\cos x},\qquad \cot x=\frac{\cos x}{\sin x}. ]

• Another identity follows directly from their definitions:

[ \tan x\cot x=1. ]

3. Special Triangles and Standard-Angle Values

The video reviews two special right triangles:

• (30^\circ!-!60^\circ!-!90^\circ): side ratio (1:\sqrt3:2), opposite (30^\circ), (60^\circ), and (90^\circ), respectively.
• (45^\circ!-!45^\circ!-!90^\circ): side ratio (1:1:\sqrt2).

From these, the key values are:

Angle (\sin) (\cos) (\tan) (\cot) (30^\circ) (1/2) (\sqrt3/2) (1/\sqrt3) (\sqrt3) (60^\circ) (\sqrt3/2) (1/2) (\sqrt3) (1/\sqrt3) (45^\circ) (1/\sqrt2) (or (\sqrt2/2)) (1/\sqrt2) (or (\sqrt2/2)) (1) (1)

The instructor also uses complementary-angle relationships, such as (\sin30^\circ=\cos60^\circ), to obtain values efficiently. When asked whether an expression is rational, simplify it and check whether any irrational quantity remains.

4. Directed Angles

Angles are assigned a sign according to their direction of rotation:

• Counterclockwise: positive
• Clockwise: negative

The order of the angle’s sides matters: reversing the initial and terminal sides can reverse the direction and therefore the sign. The instructor recommends tracing the angle from its first named ray to its second.

5. The Unit Circle

A unit circle is centered at the origin and has radius (1). For a point (P) on the circle corresponding to angle (\alpha):

[ P=(\cos\alpha,\sin\alpha),\qquad x^2+y^2=1. ]

The coordinate relationship follows by forming a right triangle inside the circle: its hypotenuse is (1), so the horizontal and vertical coordinates correspond to cosine and sine. If one coordinate is known, use (x^2+y^2=1) to find the other. When solving for a coordinate from its square, consider both positive and negative possibilities unless the point’s location determines the sign.

6. Angle-Bisector Theorems

For an internal angle bisector in a triangle, the bisector divides the opposite side in the ratio of the two sides forming the angle. If (AD) bisects (\angle A) in (\triangle ABC), with (D) on (BC):

[ \frac{BD}{DC}=\frac{AB}{AC}. ]

The video’s suggested approach is to label the two side segments, write the corresponding proportion, and solve by cross-multiplication. Perimeter information can first be used to determine a missing side or segment.

For an external angle bisector, the instructor recommends visualizing the bisector extending outside the triangle and relating the resulting segments to the triangle’s sides. The key is to identify the correct whole side and external segment before forming the proportion.

A related property is that any point on an angle bisector is equally distant from the two arms of the angle. These distances are measured along perpendiculars to the arms.

7. Medians and the Centroid

A median joins a triangle’s vertex to the midpoint of the opposite side. Therefore, it divides that side into two equal lengths.

The three medians meet at the centroid, which divides each median in a (2:1) ratio:

• Vertex to centroid: (2) parts
• Centroid to opposite side: (1) part

In an isosceles triangle, the median from the vertex between the equal sides to the base is also an altitude and an angle bisector. In an equilateral triangle, these properties coincide for each corresponding median.

8. Triangle Area

The standard formula is:

[ \text{Area}=\frac12(\text{base})(\text{height corresponding to that base}). ]

The height must be perpendicular to the selected base, or to its extension. In an obtuse triangle, the perpendicular may fall outside the triangle.

Additional area methods:

• If two triangles have the same height, their area ratio equals the ratio of their bases.
• If two triangles are similar with similarity ratio (k), their area ratio is (k^2).
• If two sides and their included angle are known:

[ \text{Area}=\frac12 ab\sin C. ]

9. Sine and Cosine Theorems

The sine theorem relates each side to the sine of its opposite angle:

[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}. ]

It is useful when a side and its opposite angle are known and another side or angle is required. The video also uses supplementary-angle facts when evaluating trigonometric values, such as (\sin(180^\circ-\theta)=\sin\theta).

The cosine theorem is:

[ a^2=b^2+c^2-2bc\cos A, ]

where (A) is the angle between sides (b) and (c), and (a) is opposite (A). It is useful when two sides and their included angle are known, or when all three sides are known and an angle is required. For an obtuse included angle, its cosine is negative, so the subtraction term effectively increases the result.

Speakers and Sources

• Speaker: One mathematics instructor/presenter, referred to as Melih in the subtitles.
• Source: Partikül Matematik, the channel identified in the video metadata.
• Euclid and Pythagoras are mentioned in passing; they are not featured speakers.