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Key ideas
Key ideas
The relationships between the length of the sides and the size of the angles in a triangle can be used to solve many problems involving position, distance, angles and area.
Different representations of the values of trigonometric relationships, such as exact or approximate, may not be equivalent to one another.
The ratio between the side length and the sine of the opposite angle in a triangle is determined by the diameter of its circumscribed circle
I can use the sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
I can explain when and how to use the sine-rule and cosine-rule to determine sides and angles in non-right-angled triangles
I can explain how to calculate the area of a triangle using trigonometric ratios
Basic Trigonometry
Basic Trigonometry
SIN COS TAN: If you have a right triangle with side lengths given and an angle (theta), you can easily find the sine, cosine, and tangent trigonometric functions. Remember that the longest side of the triangle is the 'HYPOTENUSE', the side directly opposite theta is the 'OPPOSITE', and the other side next to theta is the 'ADJACENT' side.
Once you've identified the hypotenuse (H), adjacent (A), and opposite sides (O), you can find Sin, Cos, and Tan using these trigonometric ratios: Sin of theta = Opposite over Hypotenuse (SOH) Cos of theta = Adjacent over Hypotenuse (CAH) Tan of theta = Opposite over Adjacent (TOA)
The mnemonic SOH CAH TOA can help you remember which sides to use for the sine, cosine, and tangent trig ratios. For instance, SOH stands for Sin = Opposite over Hypotenuse
Trigonometric ratios
Trigonometric ratios
Interactive
The trigonometric ratios are valuable and handy to work with because:
-> they only depend on the shape of the triangle, not on how small or large it is.
https://www.geogebra.org/material/show/id/qzhg7cfq
The sign of Sine, Cos and Tan
The sign of Sine, Cos and Tan
Discuss this table and find out the sign of cosine, sine and tangent in the various quadrants.
The Cosine Rule
The Cosine Rule
How to find angles and/or sides in non right angled triangles.
The Sine Rule
The Sine Rule
Another cool rule to find angles and sides in non right angled triangles
The Sine rule: so what do those ratios mean?
The Sine rule: so what do those ratios mean?
Let's draw a circle with a certain radius. We inscribe a triangle with one side being the diameter of the circle. The opposite angle will be 90 degrees according to Thales (semi circle theorem).
Now divide the diameter by the Sine of it's opposite (90 degree) angle. What number do you get?
Also divide the other side lengths by the sine of their opposite angle. What's the answer?
Will this also work for triangles that are not on the diameter of the circle?
Hint: Sin(90) = 1 i.e. the sine of 90 degrees is 1.
https://www.geogebra.org/material/show/id/htwkqftq
The Area of a Triangle (but not as you know it)
The Area of a Triangle (but not as you know it)
How can we find the are of a triangle when we can't find a suitable pair of base and height?
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