Trigonometric Ratios for Special Angles: 0, 45, 60, 30 Degrees
Explore trigonometric ratios for special angles like 0, 45, 60, and 30 degrees using isosceles and equilateral triangles. Understand how to calculate sine, cosine, and tangent for these angles in a clear and easy-to-follow manner.
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3 May 2025 Trigonometric ratios for special angles. LO: Find the trigonometric ratios of the following angles: 0, ?? , ?? , ?? and ?? www.mathssupport.org www.mathssupport.org
Angles 45 Look at this isosceles right-angled triangle. We let the equal sides have length 1. A Using Pythagoras theorem, the 3rd side is c2 = 12 + 12 ? ? 1 c 2 c2 = 2 c = 2 ? ? C B 1 This is an isosceles triangle, so ? = ? So, there are 2 angles each of ?? www.mathssupport.org www.mathssupport.org
Angles 45 Now look at the trigonometric ratios for this triangle Take any angle, say B. A Label the sides ? ? h opposite hypotenuse 1 2 1 sin ?? = = = 2 2 2 o ? ? adjacent hypotenuse 1 2 cos ?? = = = C B a 1 2 2 opposite adjacent =1 tan ?? = = 1 1 www.mathssupport.org www.mathssupport.org
Angles 60 and 30 Look at this equilateral triangle. Trig ratios don t depend on the size of the triangle, so we can let the sides be any convenient length. ?? ?? B ?? Let the sides be 2 cm. 2 2 You will see why 2 is a convenient length. ?? ?? 1 A 2 1 C Divide the triangle into 2 equal right angled triangles. We will take only one triangle to work www.mathssupport.org www.mathssupport.org
Angles 60 and 30 Now look at this right-angled triangle, which is half the equilateral triangle. B Using Pythagoras theorem, the 3rd side is 22 = a2 + 12 3 ?? 2 a a2 = 22 - 12 ?? a2 = 3 a = 3 A 1 So, we can calculate the trigonometric ratios for the angles ? ? ? and ? www.mathssupport.org www.mathssupport.org
Angles 60 and 30 Now look at the trigonometric ratios for this triangle We start with the angle A, ?? . B Label the sides for this angle h ?? opposite hypotenuse 3 2 sin ?? = = 3 2 o ?? a adjacent hypotenuse =1 A 1 cos ?? = 2 opposite adjacent 3 tan ?? = 1= 3 = www.mathssupport.org www.mathssupport.org
Angles 60 and 30 Now look at the trigonometric ratios for this triangle Now we take the angle B, ?? . B Label the sides for this angle h ?? =1 opposite hypotenuse 2 sin ?? = 2 3 a ?? 3 adjacent hypotenuse A cos ?? = 1 o = 2 opposite adjacent 1 3 tan ?? = = 3= 3 www.mathssupport.org www.mathssupport.org
Trigonometric ratios for special angles Summary 0 1 4 2 3 ?? ?? ?? ?? 0 0 2 1 2 2 1 4 2 3 sin 1 2 0 2 2 1 2 3 2 cos 1 0 2 3 2 1 3 ????????? tan 1 3 0 3 2 On top of the angles write down the numbers 0 - 4 Square root the number on top and divide by 2 for sine Write for cosine the same values but in inverse order Write down the angles 0, ?? , ?? , ?? , ?? . For tangent, divide the value of sine by cosine www.mathssupport.org www.mathssupport.org
Trigonometric ratios for special angles S O H C A H T O A Find the exact value of x in this triangle First label the sides We are given the side adjacent to the angle and we want to find the length of the opposite side, so we use: opposite adjacent tan = x 5 x 5 h tan ?? = o x 3 = ?? x =5 3cm a 5 m www.mathssupport.org www.mathssupport.org
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